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Wideband and low-loss negative index metamaterials for microwave antenna applications

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WIDEBAND AND LOW-LOSS NEGATIVE INDEX
METAMATERIALS FOR MICROWAVE ANTENNA
APPLICATIONS
by
David Allen Lee
Submitted in Partial Fulfillment
of the Requirements for the
Doctor of Philosophy in Materials Engineering
New Mexico Institute of Mining and Technology
Department of Materials Engineering
Socorro, New Mexico
September, 2016
ProQuest Number: 10155653
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ABSTRACT
The term negative index of refraction metamaterials (NIMs) refers to a peculiar class of engineered
materials that exhibit unusual properties not observed in conventional materials. These properties
are evident when the permittivity () and permeability () of propagating electromagnetic waves
are simultaneously negative. Because of these special properties, NIMs may become attractive
alternatives to conventional materials in microwave engineering.
Building upon fundamental research from previous work, this research synthesizes a novel design
method based upon unit cell symmetry that results in a precise transmission response; in addition,
the research introduces creative experimental methods (i.e., infrared imaging) to better
characterize electromagnetic transmission in three dimensions. This research produced prototype
materials for transmission in the X, Ku, and K microwave bands.
It is possible that metamaterials of this kind may become candidates for several specialized
applications; for example, radome materials for satellite communications require low-loss,
uniform wideband transmission for spread-spectrum modulation. In fact, many new applications
are currently being explored, and it is likely that many others have yet to be conceived that may
satisfy needs not fully met by conventional materials.
Keywords: metamaterials; microwave; negative; refraction
ACKNOWLEDGEMENTS
I gratefully acknowledge the constant support of my wife, Miriam, who encouraged my efforts to
pursue this degree during her valiant fight with cancer. Further, I thank my graduate committee:
Dr. Kalugin, Dr. Pinchuk, Dr. Majumbar and Dr. Sonnenfeld for their guidance in keeping my
progress on track throughout my graduate program. I appreciate my fellow graduate students and
coauthors James L. Vedral and David A. Smith who were essential in group discussions which
lead to creative problem solving. And I thank Professor Randall L. Musselman for his many hours
of consultation and for access to the USAFA microwave laboratory during this research. In
addition, I acknowledge three great institutions that supported this research: The MITRE
Corporation, United States Air Force Academy, and Air Force Space Command.
TABLE OF CONTENTS
LIST OF TABLES .............................................................................................................................................. 3
LIST OF FIGURES ............................................................................................................................................ 4
LIST OF ABBREVIATIONS AND SYMBOLS ....................................................................................................... 6
CHAPTER 1 INTRODUCTION .......................................................................................................................... 1
1.1 Negative Index Materials .................................................................................................................... 5
1.2 Boundary conditions for negative index materials ............................................................................. 9
CHAPTER 2 FIRST GENERATION NIMS: WIRES AND SPLIT-RING RESONATORS .......................................... 13
CHAPTER 3 ADVANCED NEGATIVE INDEX METAMATERIALS ...................................................................... 17
3.1 Design Approach ............................................................................................................................... 18
3.2 Design example for 30 GHz resonance frequency material ............................................................. 24
CHAPTER 4 EXPERIMENTAL METHODS AND RESULTS ................................................................................ 27
4.1 Experimental and theoretical results in X-Ku bands ......................................................................... 31
4.2 Experimental and theoretical results in K band materials ................................................................ 36
CHAPTER 5 INFRARED IMAGING FOR NEAR-FIELD ANALYSIS ..................................................................... 40
CHAPTER 7 CELL THICKNESS, BANDWIDTH, AND TRANSMISSION LOSS .................................................... 53
CHAPTER 8 PROPOSED APPLICATIONS ....................................................................................................... 55
8.1 Radome materials for Wideband/spread-spectrum signals ............................................................. 55
8.2 Microwave vertical/horizontal polarizer .......................................................................................... 57
8.3 Other emerging applications ............................................................................................................ 59
CONCLUSIONS ............................................................................................................................................. 61
REFERENCES ................................................................................................................................................ 63
APPENDIX .................................................................................................................................................... 66
LIST OF TABLES
Table 1. Attributes of selected split ring resonators.................................................................................. 17
Table 2. Geometrical parameters of the SSRR unit cells............................................................................ 19
Table 3 Final design fine-tuning using c (copper trace) parameter in the 30 GHz design ......................... 25
Table 4. Spread sheet format of design algorithm with intermediate values ............................................ 26
Table 5. Actual design parameters for unit cells......................................................................................... 28
Table 6. Prototype pass-bands with corresponding insertion loss ............................................................. 54
Table 7. Limited Survey of Potential Applications ...................................................................................... 60
Table 8. Template for design inputs ........................................................................................................... 66
LIST OF FIGURES
Figure 1. Positive versus negative refraction ............................................................................................... 5
Figure 2. Sketch of a prism illustrating the direction of positive and negative refraction .......................... 6
Figure 3. Transmission modes in materials by sign of permittivity (ε) and permeability (μ) ...................... 7
Figure 4. Right-hand and left-hand materials showing parallel and antiparallel behavior .......................... 9
Figure 5. Electric-field boundary condition pathway ................................................................................. 10
Figure 6. Magnetic-field boundary condition pathway ............................................................................. 11
Figure 7. Boundary conditions for (a) positive index and (b) negative index material.............................. 12
Figure 8. The modern form of split ring resonators: single SSR (top) and double SSR (bottom) with
equivalent LCR circuits on the side ............................................................................................................. 14
Figure 9. Wire mesh ................................................................................................................................... 15
Figure 10. Selected Second Generation Split Ring Resonators .................................................................. 17
Figure 11. Circuit model of S-Shaped Split Ring Resonator (adapted from Chen [22]) .............................. 18
Figure 12. HFSS produced model of SSRR with design parameters ........................................................... 20
Figure 13. Flowchart of the SSRR parameter design process ..................................................................... 22
Figure 14. Final step fine-tuning with copper trace parameter in the 30 GHz design ............................... 25
Figure 15 (left) Circuit board of 10x50 unit cells with S-shaped split-ring resonators; (top right) X-Ku
prism; (bottom right) K-band prism ............................................................................................................ 28
Figure 16. (a) the anechoic chamber (b) photo of the test setup............................................................... 29
Figure 17. Log plot of power through a NIM prism ................................................................................... 30
Figure 18. Comparison of the transmission measurements MTM (red) and HFSS theoretical simulation
(black) through the X-Ku slab...................................................................................................................... 31
Figure 19 An HFSS transmission simulation (left) at 12.5 GHz through an extended material array
transmission through a prism (right) is shown. .......................................................................................... 32
Figure 20. Transmission measurements by frequency and angle ............................................................. 33
Figure 21. Transmission measurements as a function of the angle due to a vertically polarized
electromagnetic wave (13.75 GHz) through the prism .............................................................................. 34
Figure 22. Experimentally determined indices of refraction for the X-Ku band (blue line is positive indices
and the red line is negative) ....................................................................................................................... 35
Figure 23. Simulation (black) and experimental (red) transmission through K-band ................................ 36
Figure 24. Transmission measurements as a function of the frequency and angle .................................. 37
Figure 25. Transmission as a function of the angle (21 GHz, K-band) by the metamaterial prism (peak at 40 degree) ................................................................................................................................................... 38
Figure 26. Experimentally determined IOR of the K band metamaterial: positive indices in blue; negative
indices in red ............................................................................................................................................... 39
Figure 27. The NIM prism and diagram indicating step sizes .................................................................... 41
Figure 28 Laboratory setup for IR imaging of NIMs (not to scale).............................................................. 42
Figure 29 Calibrated and processed IR image of the microwave beam of 2 Watts.................................... 43
Figure 30. High-resolution IR image (a) and (b) contour map .................................................................... 43
Figure 31. IR Images at incremental distances of 1-4 cm ........................................................................... 44
Figure 32. Simulated lobes (top) and measured lobes (bottom) for same configuration ........................... 45
Figure 33. Calibrated baseline (right) and the NIM transmission (left) ..................................................... 45
Figure 34. Image slide showing 1st and 2nd order lobes ............................................................................. 46
Figure 35. (a) Family of curves by near-field analysis showing the IOR (b) IOR by direct far-field
methods (same material) ............................................................................................................................. 47
Figure 36. Experimental set up for axial rotation ...................................................................................... 48
Figure 37. Transmission of 3-cell thick single axis as a function of axial rotation angle ............................ 49
Figure 38. Experimental Set up for lateral rotation ................................................................................... 50
Figure 39. Lateral rotation of the 3-cell thick, single axis metamaterial ................................................... 50
Figure 40. A photograph 2-D metamaterial slab (inset) for angle-dependent measurements .................. 51
Figure 41. Illustration of 1-D (left) versus 2-D design (right) transmission in rotation .............................. 52
Figure 42. Metamaterial transmission comparing cell layers of identical material .................................. 53
Figure 43. Trend in transmission loss in NIMs over the past 15 years ...................................................... 54
Figure 44. Conventional materials compared NIM designed for 20-22 GHz ............................................. 56
Figure 45. Single layer design..................................................................................................................... 57
Figure 46. Ku antenna 2.4 meter system ................................................................................................... 58
Figure 47. Ku antenna collector with up/down converter with NIM and exaggerated board spacing
overlaid to show collector .......................................................................................................................... 59
Figure 48. Flow chart for MATLAB code design tool .................................................................................. 67
LIST OF ABBREVIATIONS AND SYMBOLS
C
Capacitance
c
speed of light
cm
centimeter
dB
decibel
dBi
decibel relative to isotropic
dBm
decibel relative to milliwatt
EM
Electromagnetic
E
Electric field
FEM
Finite element method
H
Magnetic field
HFSS
High Frequency Structural Simulator
IR
Infrared
IOR
Index of refraction
L
Inductance
m
meter
mm
millimeter
MTM
Metamaterial
NIM
Negative index materials
0


degrees Celsius
electric permittivity of free space

relative electric permittivity

electric permittivity

magnetic permeability of free space

relative magnetic permeability of free space

magnetic permeability
n
index of refraction

wavelength

pi

index of refraction of a medium

velocity of waves in a medium

magnetic resonance frequency

plasma frequency
GHz
Gigahertz
MHz
Megahertz
mK
milliKelvin
NIOR
Negative Index of Refraction
SSR
Split ring resonator
SSRR
S-shaped split ring resonator
1-D
Once dimensional
2-D
Two dimensional
3-D
Three dimensional
This dissertation is accepted on behalf of the faculty
of the Institute by the following committee:
Dr. Nikolai Kalugin
______________________________________________________
Academic Advisor
Dr. Anatoliy Pinchuk
_______________________________________________________
Research Advisor
Dr. Bhaskar Majumdar
_______________________________________________________
Committee Member
Dr. Richard Sonnenfeld
_______________________________________________________
Committee Member
I release this document to New Mexico Institute of Mining and Technology
David A. Lee
02 Aug 2016
______________________________________________________
CHAPTER 1 INTRODUCTION
Metamaterials is an interdisciplinary field, and research in this area has emerged rapidly in the past
decade. The creation of the first prototype of a composite material capable of negative refraction
was preceded by several decades of interrupted efforts to realize a material based upon its earlier
theoretical development. Initially, research on metamaterials was met with skepticism by the
scientific community, but has since been validated repeatedly by several independent laboratories.
As a consequence, the vision for new engineering applications using metamaterials is expanding,
facilitated by funding from the government and commercial research. This new class of artificial
or engineered materials has emerged through interdisciplinary and international research efforts
over several decades. The focus of this research is not the physical properties of materials but the
electromagnetic properties of a novel category of materials, referred to as metamaterials and their
potential use in microwave engineering.
The term metamaterials lacks a universally accepted definition, and its use is certainly not limited
to negative refractive materials. However, the meaning of metamaterials and its place in materials
engineering has begun to coalesce. The Greek preposition ‘meta’ implies an idea beyond what is
traditionally meant. The classic textbooks usually categorized the topic of materials into these
general categories: metallurgy, ceramics, polymers, semiconductors, biomaterials, and
composites. The practice of conventional materials science is “engineering the structure of a
material to produce a predetermined set of properties”. [1] By that definition, the fundamental
material’s structure produces its properties, such as strength, inertness, heat tolerance, and strain,
under different conditions. Natural or conventional materials owe their properties to atomic and
molecular or chemical composition. In contrast, metamaterials owe their unusual electromagnetic
properties to specifically designed structures within periodic cells. Those cells are approximately
one-fifth to one-tenth the wavelength (λ/5 to (λ/10) of incident radiation. The periodic cell
structures of metamaterials influence the relative direction phase velocity and group velocity of
transmitted waves, which produce unusual properties for a particular range of incident radiation.
In short, the electromagnetic properties in metamaterials are determined by engineered structures
at the subwavelength dimensions, usually similar to millimeter scales in microwave applications.
The engineering implications of the negative refraction are of interest in many disciplines,
including optics and microwave engineering. In this research, the primary focus is microwave
applications related to wideband satellite communications. Until the present time, engineering
design methods have been considered experimental, lacking reliable methods of design tools to
achieve consistent results. Therefore, more effective and proven design methods are required to
advance microwave domain metamaterials in order to include them in the engineering trade space
in precise applications.
1
At present, research in metamaterials is published primarily in leading journals of physics,
electrical engineering, and materials engineering and sometimes in popular technologies
magazines. Several conferences are held each year dedicated to this nascent technology, which
seems to be on the verge of greater recognition as applications in electromagnetics become a
reality. Metamaterials represents a leap rather than an incremental step in materials engineering,
because it introduces an entirely new class of materials with highly unusual electromagnetic
properties. Accordingly, an expanded engineering trade space arises for microwave engineering
and optics. For example, the emergence of the microwave domain of electromagnetic
metamaterials is expected to have a transformational impact by experts in academia, industry, and
government on the engineering of antennas and related devices by designing these unusual
properties into components that enhance a combination of desired engineering traits that are not
feasible with conventional materials. [2] [3]
In the analysis of the electromagnetic (EM) interaction of matter, the properties of reflection,
transmission, and index of refraction (IOR) are largely determined by the measurement of polished
specimens at selective wavelengths. The electromagnetic parameters of permeability () and
permittivity () are generally assumed to be positive for conventional materials. Electromagnetic
metamaterials are different in these respects: 1) one or both of these parameters are assumed to be
permeability () and permittivity () negative, and 2) the response to incident EM waves is
governed by engineered frequency-scaled structures in the unit cells. In the microwave domain,
the unit cell dimensions should be much smaller than the incident wavelength (). Typically, a
homogeneous NIM requires that the unit cell dimensions be on the order of /5 to /10, where 
represents the wavelength of the incident wave. Achieving effective homogeneity within these
engineered materials is a challenge, although the objective of reducing the unit cell’s dimensions
to /10 and maintaining other design objective can be difficult to satisfy.
The negative values for permeability and permittivity may exist only for the design wavelength
range; therefore, metamaterials are inherently dispersive in nature, because an incident EM wave
at different frequency ranges behaves differently in terms of fundamental interaction. The wavematter interaction is characterized at the most fundamental level or unit cell; then, with an
understanding of the single unit, extended arrays multi-dimensions (1-D, 2-D, 3-D) are modeled
to include bulk matter interaction that includes neighboring-cell EM fields.
The fundamental structure, or unit cell, is similar to classical crystallography in some respects.
However, unlike crystals which have many permutations based upon symmetry, the unit cell in a
microwave metamaterial is comparatively simple, and its electromagnetic response is largely
determined by its permeability () and permittivity (), which can be expressed in terms of related
derivative properties (index of refraction and impedance) for the particular frequency or frequency
band. In fact, the property of negative IOR is unique to these artificial materials and do not exist
naturally. By extension, neighboring unit cells collectively determine the overall electromagnetic
properties of the metamaterial. The periodic spacing neighboring cells influence homogeneity.
As a result, new possibilities with respect to electromagnetic behavior are possible; however,
presently, design processes are not yet sufficiently mature to include metamaterials in the standard
trade space for microwave engineering due to the strict performance and tolerance requirements.
2
Electromagnetic metamaterials feature electromagnetic properties, such as transmission,
reflection, and diffraction due to the combination of ε, μ (positive/negative) values at selected
frequency ranges. Single negative parameter properties have been observed for centuries, but are
not completely understood until recently. For instance, ionosphere plasma has a negative ε below
the plasma frequency and causes atmospheric reflection. That is, the special properties of the
negative IOR (double negative parameters) only occur within designed frequency parameters.
The frequency range of interest for this research is the microwave spectrum (1-50 GHz). In this
portion of the electromagnetic spectrum, many commercial and military applications occur, such
as space-based navigation, commercial telecommunications, military communications, protected
telecommunications (including wideband spread-spectrum modulation) and others. On the
microwave spectrum, the corresponding wavelength of electromagnetic waves are centimeters or
less, which makes antennas for receivers generally small and viable as hand-held devices. As the
design processes mature and applications emerge, questions regarding metamaterials will be
answered. The goal of this research was to investigate a few important questions within the limited
scope of metamaterials; the primary focus is to design methods which yield precise designs in
terms of frequency alignment and acceptable transmission loss for a negative IOR metamaterials
and their wideband applications.
In microwave engineering, wideband signals are generated in several bands for specialized
purposes. Direct spread-spectrum modulation is implemented for secure (encrypted) and
interference tolerant communications and navigation, for example. The engineering expense of
these robust features is the necessity circuits and materials that support wideband electronics. The
definition of wideband may be vague; it is related to the signals’ modulation and is usually a
function of pseudo-random spreading codes commonly referred to as Code Division Multiple
Access (CDMA) modulation that vary considerably depending on the level of security required.
For example, a commercial CMDA system band may require a 1-2 MHz frequency band, but
support thousands of users simultaneously. Likewise, the US Global Positioning System (L-band)
uses a direct sequence spreading code of 1 Mega chip per second code, resulting in a 2 MHz band
width with unlimited users. In contrast, military-grade communications may require greater
security and resistance to EM interference; for example, frequency hopping spread spectrum
modulation may require 1-2 GHz of bandwidth and as a result are assigned to K band (18-27 GHz),
Ka band (27-40 GHz), and V band (40-75 GHz).
Based upon the theoretical foundation laid by Veselago [4] and Pendry [5], the first prototype of a
NIM was reported by D.R. Smith [6] of the University of California – San Diego. This composite
prototype, was constructed of an array of metallic resonators on dielectric boards with regularly
spaced wires, was designed to exhibit simultaneous negative permittivity and permeability in the
microwave portion of the EM spectrum. Apparently, the choice of microwave domain for this
design due to ease in construction.; however, the theory applies across the entire electromagnetic
spectrum.
Since Smith’s milestone accomplishment of designing the first prototype, there have been many
subsequent publications reporting NIMs with incremental improvements and flexibility to advance
the technology. Some of the deficiencies noted in the early prototypes were: very narrowband
response, large design-experimental errors, high transmission loss, irregular sized unit cells, and
significant inhomogeneity. This list of deficiencies may include some properties that, in fact, are
3
problematic for every microwave application, but in general, the items on the list below are
considered undesirable and have been noted as barriers for most practical engineering applications.
Issues with early prototypes of NIMs:





High loss
Narrowband Transmission
Substantial model-experimental errors
Inhomogeneity
Polarization sensitivity
Although the phenomenology of negative index had been observed, as predicted by Veselago, the
design process was immature, leading to large variations, suggesting an incomplete engineering
process which would be needed to reliably design devices precisely. Furthermore, scaling and
tuning materials to frequencies within a band is instructive in order to appreciate which design
parameters are most appropriate to optimize product development. The issues to be addressed for
continuing research are whether these deficiencies were associated with first-generation designs
or whether subsequent designs based upon the same theory could be made to mitigate deficiencies
to make the new technology more attractive for its use.
Currently, relatively mature NIM design processes lack the precision required for very demanding
applications such as satellite communications. Previous research indicates a substantial variation
in modeled versus experimental results. For example, in designing a material for design band-pass
regions, considerable errors in center frequency and bandwidth are often found; errors up to 24%
are found compared to the desired design center frequency. [7] For the most stringent application,
product design and prototype discrepancies are problematic. Conventional components such as
analog filters, for example, must perform with high precision for most communication circuits.
Accordingly, advanced design tools are needed to produce good design-experimental agreement.
As design methods mature, the design process is expected to yield a clearer understanding of
scaling across the spectrum.
Research questions considered:







Can the models be applied across the microwave spectrum?
Can low-loss/wide band materials be achieved with innovative design?
What unit cell thickness is required for practical materials?
Is near-field transmission performance similar to far-field?
Can the sensitivity to polarization be neutralized or exploited?
What applications show promise?
Can measurement and analysis be expanded to 2 and 3 dimensions?
In order to address the first question on model fidelity/maturity, a thorough literature search was
performed. The findings of the search revealed that insufficient attention had been paid to
resolving model discrepancies and experimental results. If metamaterials are to be seriously
considered for critical industrial applications, reliable engineering methods must be developed to
curtail extended iterative, slow-converging designs approaches. Furthermore, these designs
should yield products that suit a variety of applications.
4
1.1 Negative Index Materials
In 1968, Victor Veselago predicted that materials with simultaneous negative values of electric
permittivity (ε) and magnetic permeability (μ) could exist based upon fundamental laws of
electromagnetism. [4] He also noted that such dispersive materials were unknown in nature. After
this publication, his work did not draw much attention perhaps because there was not a clear
pathway to engineer these materials; nonetheless, his work laid a theoretical framework. Through
Maxwell’s equations, Veselago explained how some properties of propagation change while being
transmitted through NIMs; he also referred to these materials as “left-hand materials”. For
example, NIMs are dispersive and demonstrate a reversal in Snell’s law and boundary conditions
as well as antiparallel relationships of phase velocity for backward propagation.
In conventional materials, the assumption is that the material’s parameters (ε, μ) are real and
positive; however, in the general case for index of refraction (n) in Eq. 1 and Snell’s law in Eq. 2,
and both positive and negative indices of refraction are possible and are illustrated in Figure 1.
 = √  (1)
Medium 1
Medium 2
Positive
Refraction
Negative
Refraction
Figure 1. Positive versus negative refraction
  =   (2)
A negative refractive index means that the phase of the transmitted wave delays instead of
advances through the medium; that is, the phase velocity of the wave may be either positive or
negative as in Eq. 3.
 = √  (3)
Furthermore, since the IOR is similarly defined as the ratio of the velocity of light (c) in a vacuum
to its velocity through a particular material ( ), as in Eq. 4.
 = / (4)
5
As the EM waves are transmitted through the NIM, the transmission vector is on the negative side
(labeled in Figure 2) of the prism’s normal line; otherwise, the transmission vector is dominant on
the positive side.
Figure 2. Sketch of a prism illustrating the direction of positive and negative refraction
To illustrate further, a diagram is constructed in Figure 3 to illustrate how the sign
(positive/negative) of the parameters affects the nature of wave transmission in different categories
of media. The figure is arranged with the horizontal axis representing the signed value
(positive/negative) of the electric real ε and μ of materials, and the types of electromagnetic waves
supported in each quadrant. This graphic is divided into four quadrants as created by the sign of
each parameter. When the sign of both (ε and μ) parameters are equal (both positive or both
negative), transmitted waves are propagated as indicated; when signs are unmatched, waves are
evanescent or rapidly decaying. The upper right hand (RH), indicates forward propagation, and
lower left hand (LH) indicates backwards propagation. These modes are labeled in Figure 3 as
opposite quadrants I and III and indicate forward and backward propagation, respectively. In
contrast, opposite quadrants II and IV indicate mixed signed (+/-) parameters which result in
evanescent (rapidly decaying or vanishing) waves; materials in the domain of quadrants II and IV
shall not be discussed further in this thesis because this research if concerned primarily with media
that supports propagation.
6
+m
Forward (RH)
propagating waves
E
Evanescent
waves
Optical (l)
S
k
-e
H
II
E
k
Backward (LH)
propagating waves
I
+e
Evanescent
waves
Non-Optical (l)
S
H
k
 k IV
III
-m
Figure 3. Transmission modes in materials by sign of permittivity (ε) and permeability (μ)
As a result of these LH materials, many new kinds of optical lenses are possible (for example,
phenomena such as a reverse Doppler effect, which does not have any practical applications. Also,
NIMs manifests a reversal of Snell’s law and boundary conditions as well as antiparallel relations
of phase velocity for backward propagation.
In summary, Veselago’s criteria for left-hand materials are:








Dispersion of constituent parameters (,)
Reversal of Snell’s Law
Reversal of Boundary Conditions
Negative Refraction
Exchange properties (i.e. convergence and divergence) of convex and concave lenses
Issues with isotropic property
E field, H field
E x H = S are constant direction, but wave propagation (k) is ‘antiparallel’
Like the IOR, impedance in Eq. 5 is a complementary metric also derived from a relationship
between permeability and permittivity but is more common in electrical engineering, especially as
applied in microwave technology.
()

 = √ () =  (5)
7
The orientation of the pointing vector (S) is consistent, regardless of the phase velocity vector, as
shown by Eq. 6.
 =  ×  (6)
Maxwell’s equations are the basis for electromagnetic properties: magnetic and electric fields in
free space and their properties at the interface of material (boundary conditions) as the governing
set of equations for electromagnetic behavior.
Following the notation used in Veselago’s paper, the differential forms for Faraday’s Law and

Ampere-Maxwell’s Law ( ⃗ = 0) and substituting  = . It follows that Maxwell’s relations for
electric and magnetic fields in LH Materials are shown in Eq. 7 and 7’ and Eq. 8 and 8’.
⃗⃗

⃗⃗⃗⃗
∇ x ⃗⃗ = - 
(7)
 ×  = − (7’)
⃗⃗⃗⃗
⃗⃗ =  ⃗ −  
∇x


,  ⃗ → 0
(8)
 ×  =  (8’)
Now, substituting for the propagation vector, we apply the propagation vector (k) as in Eq.
9 and Eq. 10.
 ×  =  (9)
 ×  = − (10)
Taking the absolute value of the parameters, we have a revised set of equations as in Eq.
11 and Eq. 12.
 ×  = −|| (11)
 ×  = || (12)
⃗⃗ )
The pointing vector has a right-handed form of triple vectors: Electric (⃗⃗ ) , Magnetic (
⃗⃗ , where the group velocity ⃗⃗⃗⃗⃗⃗
and 
 is always in the same direction of ⃗ (energy
⃗⃗ , when parallel (right handed) is same direction as ⃗, but antiparallel
propagation) and 
(left handed) involves vectors pointing in a different direction in the left-handed form, as
seen in Figure 4.
8
Figure 4. Right-hand and left-hand materials showing parallel and antiparallel behavior
1.2 Boundary conditions for negative index materials
As electromagnetic waves cross the interface of two media, they are subject to boundary conditions
that are derived from Maxwell’s equations. As such, these boundary conditions are equally valid
for right-hand media and left-hand media and apply to time-dependent and time-independent
fields.
The final objective in this section is to examine the behavior of electromagnetic waves at the
interface of two dielectric media: one conventional right-handed dielectric and the second a set of
left-handed metamaterials. Initially, the derivation of common boundary condition is reviewed,
and the distinctions on how electromagnetic transmission occurs through metamaterials will be
covered.
Prior to the examination, a few assumptions about the nature of the media being considered must
be made. If the dielectric is anisotropic, D, E, and P are not parallel. In the Cartesian coordinate
system, the permittivity is different along each axis, as shown in the matrix in Eq. 13.


[ ] = [





 
 ] [ ] (13)
 
In the general case, the electric flux density value depends on the orientation (i.e.,  =   +
  +   ) in the x direction. In computational electromagnetics, calculations of this type
of matrices are used to estimate the electric flux density based upon electric fields. For simplicity,
an isotropic media is assumed for the purposes of analysis of boundary conditions. Instead of an
isotropic media, an effectively homogeneous media is assumed. A truly homogeneous material is
linear in which Eq. 14 strictly applies. An effectively homogeneous media approximates linear
behavior with acceptable errors between model and experimental measurements.
9
 =  (14)
For the derivation of electric-field boundary conditions, we begin with the integral form of
Faraday’s law. Eq. 15 describes the entire integral path across the boundary interface, and Eq.
16 provides a more detailed equation illustrating that each step in the integral pathway is broken
down, when matched to the pathway parameters as depicted in Figure 5.
⃗⃗= 0 (15)
∮ ⃗⃗ ∙ 




∫  ∙  + ∫  ∙  + ∫  ∙  + ∫  ∙  = 0 (16)
Figure 5. Electric-field boundary condition pathway
To simplify the sum of integrals in Eq. 16, the dimension ∆ℎ is allowed to approach zero: ∆ℎ →
0 while the ∆ dimension remains static. As a result, Eq. 16 is simplified to eliminate two
normal components of the E-field and leave only the equivalent tangential components: 1 ,
2 , with subscripts indicating the tangential component (t) and media (1) and media (2) as
shown in Eq. 18.


∫  ∙  + ∫  ∙  = 0 (17)
1 = 2 (18)
For the derivation of magnetic-field boundary conditions, beginning with the integral
mathematical form of Gauss’s law for magnetic fields in Eq. 19.
⃗⃗ ∙ ⃗=  (19)
Eq. 1 ∮ 
10
The magnetic flux from each surface (top, bottom, and side) of a cylinder is shown in Figure 6,
where a function (difference) of only the normal components of (2 −1 ) is presented, as in Eq.
20 and 21.
Figure 6. Magnetic-field boundary condition pathway
∮  ∙  = ∫  ∙  + ∫  ∙  + ∫  ∙  (20)
( − ) ∙  =  (21)
Therefore, the components of the electric field follow the cardinal rules of the EM boundary
conditions; that is, the tangential components of the first media 1 equals the tangential
components of the second 1 ; 1 − 2 = 0 ; likewise, the normal component of 1 − 2 =
 and where  = 0 as in radiation in free space, then 1 − 2 = 0, which can be represented
as 1 1 = 2 2 . What is apparent from the diagram is that the sign of the tangential component
is the same, but the sign of the normal component is different.
To illustrate the behavior of an electromagnetic wave incident upon a NIM, it is instructive to
compare (side by side) diagrams of both positive index and negative index materials. In Figure 7,
an electromagnetic wave propagates from medium 1 (top) into medium 2 (bottom) at an angle
 with respect to interface normal; a portion of the incident wave is reflected at the sample angle
but labeled as  (distinguish the two) and transmitting through media 2 and an angle  , according
to the IOR in Snell’s law. For the positive index material Figure 7 (a), the propagation vector (k)
and Poynting vector (S) are consistent in both media. In addition, the tangential component of 1
(1 ) equals the tangential component of 2 (1 ) at the boundary, as discussed earlier in this
section. The normal components of the wave behave according to the relation: 1 1 = 2 2 .
Accordingly, unless the indexes of the refraction of the two media are identical, the normal
component (2 ) differs by the ratio 1 ⁄2 , and since the normal component of the vector changes
across the boundary, all associated vectors 2 , 2 , 2, 2, change in direction and magnitude.
11
What seems obvious in the side-by-side comparison of positive versus negative index materials is
that the sign of the normal component of the E-Field changes from positive to negative as it crosses
from media 1 to 2. Likewise, the direction of wave vector 2 reverses. Still aligned in parallel
with 2 , which carries energy and modulation, 2 points in the opposite direction as shown in
Figure 7 (b).
(a)
(b)
Figure 7. Boundary conditions for (a) positive index and (b) negative index material
12
CHAPTER 2 FIRST GENERATION NIMS: WIRES AND SPLIT-RING RESONATORS
Nearly thirty years after Veselago published his seminal paper, renewed interest in Veselago’s
theories was taken by Professor John Pendry (Imperial College,UK), who investigated unique
devices to realize simultaneous negative parameters (permittivity and permeability). In a number
of publications, Pendry derived models for effective permittivity ( ) and permeability ( )
by engineered cells in materials that produce negative index values. Pendry notes, “every material
is a composite, even if the individual ingredients consist of atoms and molecules. The original
objective in defining permittivity and permeability was to present a homogeneous view of the
electromagnetic properties of a medium. Therefore, it is only a small step to replace the atoms of
the original concept with a structure on a larger scale. Periodic structures were defined by an array
of unit cells and of characteristic dimension a. The contents of the cell define the effective response
of the system as a whole.” [8] As a theoretician, Pendry investigated the physics combining the
electromagnetic properties of wire meshes and split-ring resonators to produce negative values for
 and  however he did not pursue an experimental solution.
Properly designed split-ring resonators (SRR) that result in the negative permeability for a selected
frequency range while a wire mesh with specially designed geometry and spacing results the
negative permittivity for an overlapping frequency range. The overlapping frequency range is
considered simultaneous negative. The SRR resonant frequency  = 1⁄√ is a characteristic
frequency in which the negative index is anchored. If mutual coupling is weak, the double-ring SRR
configuration has about the same resonant frequency as long as the physical dimensions are similar and
electrical components are roughly equal: 1 ≈ 2 , 1 ≈ 2. The rationale for a double SSR is to produce a
large magnetic moment, created by higher current density SRR effective permeability, as follows in Eq.
22.

µ() =  −  −
 + +
(22)
Where:
 = (⁄)2 ;
 = √3⁄ln(23 /), where  = spacing between rings
 = 2 ′ / ; R’ = metal resistance/length, experimentally derived
To achieve an effective homogeneous material, Pendry concluded that the dimension (a) must unit
be smaller than the incident wavelength (l); otherwise, the internal structure of the material might
diffract the incident radiation. For completeness, Pendry’s work was built upon the contributions
of several predecessors. His work was preceded by Hardy’s related work [9] in the 1980s that had
produced a similar structure that exhibited resonance and coined the term ‘split-ring resonator’ for
13
the design frequency of about 1 GHz. Furthermore, D.L. Mills [10] and R.E. Camley [11]
contributed extensively to the pertinent research on metallic properties of negative permeability
() by offering new scientific insight into the electromagnetic response with arrays of SRRs.
Nonetheless, Pendry and his coworkers are generally credited for the modern form of the split ring
resonator, which is a pair of concentric split rings with opposing splits and therefore opposing
currents. The current is mostly confined to the outer perimeter as shown in Figure 8, with the Efield and H-Field and propagation vector (k) included with the SSR and equivalent LCR circuits
shown side by side.
L
C
E
R
. k
H
L1
L2
C1
R2
C2
R1
Cm
Lm
Figure 8. The modern form of split ring resonators: single SSR (top) and double SSR
(bottom) with equivalent LCR circuits on the side
The ring is a comparable structure used as a building block with neighboring SSRs to create a
homogenous slab of negative index metamaterials. In effect, the magnetic field is polarized normal
to the surface of the SSR, caused by an oscillating electric field in parallel to the plane. The plate
is slightly diamagnetic. The EM response of a single split ring is purely inductive; however, when
a smaller split ring is introduced inside the outer ring with a gap (d) with sufficient spacing and
capacitance (C) is provided to construct a resonant circuit. When the splits are opposed (the outer
split is 180 degrees out from inner split), capacitance is enhanced. The perimeter of the split ring
should be well below the dimensions of the wavelength of the radiating wave. Without a double
ring, charge accumulates near the slits and creates an undesirable dipole moment.
The double ring split-ring circuit is essentially a LC equivalent circuits in response to the electric
field; the ring forms and inductive component with the value of 2  . While the capacitance is
a function of the inter-ring space (d) and the split distance, the capacitance between the rings is the
dominant contribution. The value  ≈ 2 ( + )/d is evaluated for resonance and total C
=  /4. Neglecting ohmic losses and collecting those terms, to model a LC resonant circuit with
the resonant frequency is  = √1/() .
14
The term plasma frequency is a common parameter to describe a network of thin wires that have
electromagnetic properties much like low-density plasma, when the electric field is parallel to the
axis of the wires. The wire network has the property of negative emissivity below the plasma
frequency ( ), while the permittivity () and simplified effective permittivity (adopted by
Smith) is shown in Equation 14. Other contributors include R.N. Bracewell [12] and W. Rotman
[13], who investigated the EM properties of wire arrays, concluding that wire arrays can be
designed to generate negative permittivity () for frequency ranges below the plasma frequency.
In bulk, wire arrays demonstrate similarly to neutral plasmas in the ionosphere. The equation for
the plasma frequency of a wire mesh ( ) is shown in Eq. 23, where c is the velocity of light and
the dimension a defines the unit cell lattice while an r is a wire’s physical radius as shown in
Figure 9.
 =
 
(23)


 ( )
Figure 9. Wire mesh
Having a means to design a plasma frequency based upon wire mesh, we can also extend that
property to reflect its permittivity () in Eq. 24 and effective permittivity  () in Eq. 25,
which is a simplified form that assumes that damping ( ) is negligible.
+ 

() =  −  − +
(24)


 () =  −  (25)
Armed with a proven model for the negative parameter  () , which occurs when (2 >
2 ) , a research team at the University of California – San Diego that explored Pendry’s
theory for devices with negative permeability. The team envisioned a composite material
in which both negative parameters were simultaneous (  ,  ).
15
Smith and his team successfully designed, fabricated, and tested a NIM prototype, tuned
for the microwave region in one dimension. [6] However, having proved the concept in
uncoordinated space, the validation can be extended to higher-order dimensions 2-D and
3-D if the material can be made effectively homogeneous, the fundamental design criteria
being unit cells (containing SRR and wire mesh arrays) with dimensions on the order of
 < l⁄8 , satisfying some standard of symmetry with plane wave orientation. In testing,
symmetry is necessary to ensure that the transmission is insensitive to the orientation of
the incident radiation.
After Smith’s initial work was published, a period of great interest followed. Some
researchers questioned the conclusions, claiming that the experiments and analysis were
flawed. [14] [15] However, over a period of several years, the realization of the negative
IOR was confirmed by several independent researchers. [16] [17] [18] [19] After the
realization of this new class of materials was generally accepted, greater attention was
given to addressing potential improvements with certain application in mind. As a result,
the development of the next generation of negative index materials was considered by a
large community of international pure and applied researchers.
The objective of some of these researchers interested in microwave applications was to
develop materials with sufficient transmission bandwidth to support modulated signals
with acceptable loss. Successful efforts were reported in transmitting microwave signals
using modulation techniques, such as Quadrature Phase Shift Keying (QPSK) [20];
however, the decoding mechanisms failed when transmitted through metamaterials at
higher bandwidth signals. Although not explicitly known, the non-uniformity of wideband
transmission is clearly problematic. In modern space-based communications, specialized
modulation techniques include spread-spectrum modulation with sacrifices using large
bandwidth for corresponding tracking robustness in electromagnetic interference. For
these specialized cases, efficient transmission over very large bandwidths (up to 2 GHz) is
desired. To accomplish uniform low-loss transmission, novel materials must be
considered, since conventional material design simply is ill suited for this combination of
properties.
16
CHAPTER 3 ADVANCED NEGATIVE INDEX METAMATERIALS
As of this writing, there are several different design families of metamaterials based upon
the SRR; some of these were summarized by Engheta et al. [21] Figure 10 illustrates four
examples of SRRs which have emerged as promising candidates for microwave
applications. In the figure, elements associated with negative permittivity are red, and those
with negative permeability are light blue, with the exception of the S-Shaped Split Ring
Resonator (SSRR), which is unusual in that it has no separate rods.
Figure 10. Selected Second Generation Split Ring Resonators
The properties of this selected group are shown in Table 1. The unique S-Shaped SRR
(SSRR) unit cell architecture provides both negative parameters in a simple, symmetric
design that lends its performance to wide-band, low-loss transmission performance, by
avoiding rods which are associated with high loss transmission and fabrication issues.
Split Ring Resonator
Type
Axially-symmetric
Edge-coupled
Omega
S-Shaped
Properties
1)
2)
1)
2)
1)
2)
1)
2)
3)
4)
5)
Fair Transmission
Rod issues
Transmission issues
Contact rod issues
Contact rod issues
Cancels bianisotrpy
Excellent Transition
No rods
Wideband (GHz)
Ease of fabrication
Symmetric
Table 1. Attributes of selected split ring resonators
17
In reviewing the current set of approaches, the choice made seemed an interesting
candidate to develop because of its wide bandwidth property, which may prove useful in
spread-spectrum communications, for example. As a result, it is selected here as the
fundamental unit cell for negative refraction. A comprehensive review of the SSRR device
theory is perhaps best explained in two papers by Chen. [22] [23]
Figure 11, adapted from Chen, illustrates the capacitive and inductive components derived
from the SSRR that enable the resonant circuit behavior. Figure 11(a) depicts the circuit
components as the capacitance between parallel metal strips and (b) shows the equivalent
LC circuits with current (j).
(a)
(b)
Figure 11. Circuit model of S-Shaped Split Ring Resonator (adapted from Chen [22])
3.1 Design Approach
In the design of circuits for communications, especially space communications,
performance parameters such as center frequency, bandwidth, and roll-off are demanding.
Thus, if metamaterials are to be considered as engineering tools, design methods must
produce precise response according to requirements. To date, that level of precision has
not been available in the design of NIMs, contributing to the reluctance by the engineering
community to include these materials in the engineering trade space.
In general, the design requires very precise control of the unit cell design. In this study, a
50-micron resolution was the limit of the design tools available. Industrial resolution
continues to push the state of the art, and accordingly, 10-micron resolution may soon be
the industrial standard for chemically etched printed circuit boards.
18
Once the design tool resolution is determined, the next step is cell size approximation. In
theory, a homogenous material should have a unit cell size; that is, its largest dimension
should be about /10 or less. In practice, this is difficult to achieve, and a unit cell size
between /5 to /10 may have to suffice. In the S-shaped split-ring resonator unit cell,
there are six essential dimensions that define the dual S-shapes (front and back), in addition
to a seventh (l) parameter, which indicated the distance between unit cells, as Table 2.
SSRR
dimension
a
b
h
w
c
d
l
Description of design parameters
Vertical cell perimeter
Horizontal cell perimeter
Vertical S-Shape
Horizontal S-Shape
S-Shape line thickness
Front-back board thickness
Board-to-board distance
Table 2. Geometrical parameters of the SSRR unit cells
These seven design parameters coupled with the choice of dielectric and metal materials
are the complete engineering trade space. If one elects to use commercially available
laminates for microwave frequencies, then a limited set of the laminate thickness (d) is a
restriction, as is the default choice of copper (1 or ½ oz.) and dielectric materials. However,
microwave engineering faces relatively low cost of such materials and abundance of
industrial design tools, such as Ansoft’s finite element method simulation product, High
Frequency Structural Simulator (HFSS), which has integrated libraries of commercial
products and options. These design parameters are shown in the illustration below, shown
in Figure 12 perspective (a) and frontal (b) aspects, to fully illustrate the design parameters.
⃗⃗⃗ and 
⃗⃗⃗⃗ are
In addition, to those design parameters, the electromagnetic vector fields for 
in optimal alignment; the orthogonal electromagnetic vectors govern the direction of
⃗⃗).
propagation (
19
l
d
(a)
(b)
Figure 12. HFSS produced model of SSRR with design parameters
Using these design parameters, it is possible to select a set of values to initialize simulations which
converge on the desired electromagnetic properties. However, the key to rapid convergence in the
design process is to select the parameters in an orderly sequence based upon coarse to fine
sensitivity; that is, choosing first the most sensitive design parameter in which small incremental
changes translate into very large changes in terms of resonant frequency to get an approximate
performance within the possible range. Once the designer is satisfied through the modeled
simulation that a reasonable approximation is achieved by adjusting that single parameter, less
sensitive parameters can be adjusted in order.
The challenge, of course, is to learn the sensitivity order to which these parameters (listed in Table
2) belong. Without that knowledge, the designer may spend unnecessary time and effort in design
iterations before a convergence is found. It should be stated that design experience shall reduce
effort by introducing better initial parameters with materials selection; consequently, given the
extensive library of materials available in commercially available finite element method (FEM)
tools, the preliminary model should yield a close approximation of the design parameters to pass
on the FEM simulation. Other limitations may be imposed by the machining precision tolerance
and circuit board metal traces, which are necessary details to be included in the FEM simulation.
Therefore, modification to the FEM design tends to be very minor, prior to generating a final
GERBER (industry standard) file for circuit board fabrication.
Expeditious use of the FEM tool as a final step may be important if the designer is constrained by
the time-use of a community license on a shared server. Most corporations and universities limit
access time to these tools because of limited seats and high demand. If fact, access to these FEM
tools is of critical consideration, especially when extended arrays (prisms and slabs) are being
simulated. In reality, it may not be feasible to simulate an extended model of many unit cells for
a large material without sufficient time on a server with rigid time limitations. In this research,
progress was linked to the availability of FEM tools and progressed nicely when the tools were
migrated to a dedicated server provided by the US Air Force Academy, which then implemented
FEM tools using an efficient parallel processor with only a few users.
20
The flow chart below shows the sequence and the rationale for each step, in the sequence which
reduces the iterations necessary to meet the design frequency. As previewed, the board thickness
(d) is a discrete amount – usually 1 or 2 mils, which was clad with 1 oz. copper. After choosing
between those options, board spacing is the logical next step because of the sensitivity; single steps
in the spacing (typically 1mm increments) translate into large offsets in frequency. With those
steps assumed, the resonant frequency should be approximated through the unit cell size and
internal to the cell of the S geometry. The cell size is determined by the product of a (height) and
b (width). If both upper and lower cells are equal, as was true in this case, the estimate is simplified
and inherently more homogenous to incident EM waves, which is highly desirable. The frequency
is proportional to ab/2. When appropriate parameters are calculated, they should be compared to
the range of /5 and /10; the effective homogenous rules stipulate that the first is necessary
and the second is the objective, and there is no clear break point in between. In fact, the hypotenuse
of components a and b should be near their minimum.
An algorithm was created using MATLAB script and executed to simulate resonance response
based upon the (real part) permeability. In addition, design constraints were imposed on the unit
cell to improve homogeneity of the bulk material. As described, design constraints provide
symmetry and of the active area in the cell of 0.7 or less. The bulk material was designed to reduce
diffraction effects.
In resonant circuits and materials, the magnetic resonance frequency  is governed by the
square root of the reciprocal product of inductance (L) and capacitance (C). The specific
relationship for the SSRR is shown in Eq. 26.



 = √ ( +  ) (26)



The inductance L of each circuit (half ring of S-Shape) is proportional the enclosure area, as in Eq.
27. Clearly, the area of the half ring (ab/2) is a key design parameter that determines the size of
the unit cell. Higher frequency unit cells are correspondingly smaller in size.
 = 
(27)
In addition, the capacitance created ( top/bottom of S-shape and  middle of S-shape) by
matching metallic strips (h*c = metal area) separated by board thickness () of dielectric material
and by neighboring boards separated by free space at a distance ( − ) may be calculated by Eq.
28.


 =  =   (  ) +  (−) (28)
Again, the parameters a, b, h, c, d, and l are the geometrical parameters shown in Figure 12. In
addition,  is the resistance of the metallic strips in each loop and  represents the fractional
area.
The flowchart in Figure 13 illustrates the design process, beginning with the user input of a single
cell design frequency, frequency error tolerance, SSRR parameters, wavelength cell parameter,
etc. The outcome convergences on the input parameters, generally, in a few steps.
21
Figure 13. Flowchart of the SSRR parameter design process
The magnetic permeability the SSRR is a function of these design parameters, which are modeled
by series of equations, including Eq. 29 for effective permeability are found in Chen’s paper. [23]
 =  −
+


− 
( )–+
   
(29)
where the collection of some terms is:
22
=
=
2     2   

)
      
(   )2 (1−
()2
( )2 (1−
=

)
 
22
)
 

( )2 (1−
)
 
()(2 −
,
Although this set of equations provides a theoretical means to design a NIM, there are important
pragmatic considerations that must be understood before the modeled performance is realized in
the laboratory. If the designer intends to use commercial circuit board technology to fabricate the
NIM, pragmatic considerations limit the raw materials (metals, dielectric values thickness
dimensions, etc.). Familiarity with those parameters up front expedites the design process by
eliminating unavailable design choices; that is, since nearly continuous design options are
available in modeling, discrete increments for board material thickness, for example, industrial
standards may offer only certain dimensions.
The challenge in this process is the selection of parameters to modify to optimally converge on the
final design. Beginning this research, no systematic design approach was reported in the literature.
Without an approach to reduce errors, it is unlikely that the special properties of metamaterials
will be considered for industrial applications where precision is required. To achieve a design
process that converges necessarily, an accurate sensitivity analysis of each of these parameters is
needed. In general, the most sensitive parameters (those that influence performance significantly
with small changes) should be addressed first. Once the order of the sensitivity is understood, the
proper design sequence leads to rapid convergence to an acceptable design. A successful design
process is one in which design decisions are by order; rank of parameter sensitivity order may be
a practical matter when choosing materials fabricated by commercial circuit boards.
The designer must be aware of constraints imposed by industrial circuit board products and
constraints by milling equipment; in this case, options were 1 mm and 2 mm spacing, where
predetermined resolutions elected via discussions with machinists. In addition, the available
options for board thickness (d) may be quite limited to discrete multiples (mils), such as 5, 10, and
20 mils. For example, 20 mils (0.508 mm) was the design in this work, leveraging data from
successful trials. Similar constraints may apply to other designers in choosing major
configurations to approximate the frequency range in order to fine tune the resonance frequency,
the details of the SSRR, and the width of the copper trace (c).
Unit cell size is determined by the area (ad). The literature reports many experimental designs,
which are asymmetric (a ≠ b), that a design approach may work well for the 1-D material (E
polarized always aligned) with the resonator. However, when materials are expected to perform
similarly when the polarization of incident radiation is unknown, symmetric cells (a=b) behave in
a more homogenous manner. Likewise, the inner dimension (h=w) of the metallic material
contributes to a homogenous performance.
23
In addition, geometrical constraints were imposed by unit cell design on the active (copper)
portion. The unit cell dimensions, spacing and board thickness are key design considerations. For
this research, 20 mil (0.508 mm) thick commercially-available circuit board were selected for use
in the final designs.
The initial estimate may need tuning of center frequency, which leads to several options in terms
of parameter modification. Since the permeability governs frequency resonance (assuming
permittivity is negative), then the permeability must be manipulated to achieve the target frequency
response. In simulation, given these seven parameters, the designer must make a first estimate,
then simulate the response to check the response and then modify at least the parameters to shift
in the direction to close the difference. For example, if the desired performance is to respond as a
band-pass, the center frequency and bandwidth are approximated in an initial simulation.
In modeling, the real component of theoretical permeability () is used to model the resonant
frequency of the material, making an assumption (at this point of that stage) that the real
permittivity () at the same frequency is negative. This usually (but not always) proves to be a fair
assumption but shall have to be verified in the later FEM stage of the simulation. If the assumption
of a negative permittivity is flawed or the FEM modeled permeability does not align with
expectations, revisions shall be necessary. For instance, issues with earlier prototypes (not
reported) produced theory-experimental discrepancies that were not well understood until higherquality materials with thinner board thickness were used. Thereafter, better theoretical and
experimental agreement results are consistent.
3.2 Design example for 30 GHz resonance frequency material
As a design example of the design process, the design frequency is selected at 30 GHz, using the
same material (Rodgers Duroid) and board thickness (d) of 1 mil (0.508 mm) as well as the design
parameter l shall be made in 1 mm increments. To approximate the resonant frequency of 30 GHz,
the unit cell governed by the area (ab) parameters should be on the order of /5, and the active
portion of the cell (hw) should be about 0.7 (empirically derived) per side of the unit cell area.
The wavelength () at 30 GHz is 10 mm. In order to achieve /5 for the active portion, the a and
b dimensions is 2 mm on each, and the active h and w unit cell areas should be (0.49) of the ab
area. Table 3 contains the set of initial design parameters, with the first column (l) being used as
the first variable to select the board spacing most appropriate for the 30 GHz design. The choice
of 0.9 mm for the copper trace thickness is an approximate estimate, since it is usually reserved
for fine-tuning during the final design step. In addition, Table 3 indicates the design variables
being considered; in the last column, the simulated resonant frequency is listed corresponding to
the board spacing. In this case, the board spacing of the l = 1 mm leads us to the best approximation
of desired resonance frequency response, as shown also in Figure 14.
24
L
d
1
0.508
SSRR Design Parameters (mm)
a
b
h
w
2
2
1.5
1.5
c
Cell size
n ( l/n)
tolerance
(GHz)
0.488
5
0.025
Design Sim. Freq.
GHz
(GHz)
 30 +/- 
30.02
Table 3 Final design fine-tuning using c (copper trace) parameter in the 30 GHz design
Figure 14. Final step fine-tuning with copper trace parameter in the 30 GHz design
The final step in this design is to adjust the copper trace width, ‘c’ parameter; this design step finetunes of the resonant circuit, as it can be modified by very small increments (10 micron or smaller)
according the circuit board technology. In this design example, the design parameters converged
within 5 steps, using the ‘c’ parameter (copper trace thickness) as the fine-tuning parameter to
meet the frequency tolerance stipulated.
25
For this example, it may be instructive to break down the results of the algorithm, using a spread
sheet format to indicate intermediate values with corresponding units. This result is shown in
Table 4. Some of the design parameters prompt the user to select a particular range of values as
inputs; in addition, the computational process is simplified where possible due to symmetric
parameters.
Symbol
Design Parameter
f
Design Freq
n
Cell size (lamda/n) n=5:10
d
Board thickness
l
Spacing (between boards)
Er
Dielectric constant (4003C)
a
Unit cell High
b
Unit cell Wide
h
SSRR High
w
SSRR Wide
x
Fraction (h/a), (w/b) x ~ 0.75
C
Capacitance for case (Cs = Cm)
F
Area Fraction (wh/ab)/2 for case [F1 = F2]
S
Area Unit Cell (a*b)
epsilon(o)
epsilon free space
mu(o)
mu free space
c
Copper Trace (fine tune here)
Fmo'
Magnetic Res. Freq.
Value
30
5
5.08E-04
1.00E-03
3.65
2.00E-03
2.00E-03
1.50E-03
1.50E-03
0.75
5.97E-14
2.81E-01
4.00E-06
8.85E-12
1.26E-06
4.88E-04
30.003
units
GHz
unitless
m
m
unitless
m
m
m
m
unitless
F
unitless
m^2
F/m
H/m
m
GHz
Table 4. Spread sheet format of design algorithm with intermediate values
The MATLAB version of the algorithm is dynamic, incorporating a proportional feedback loop
intended for rapid convergence on the solutions. It also interpolated between frequency maxima
(positive and negative) near the transition to provide a better approximation. However, the static
spread sheet method enables the user to inspect intermediate values and develop an intuitive sense
of sensitivity for parameter changes. Both versions are useful, but the MATLAB version is
preferred in generating design parameters to transfer to HFSS modeling in the final step.
Once the design algorithm is run through convergence within the specified magnetic resonance
frequency tolerance, the design is further fine-tuned, adjusting only the copper trace parameter to
achieve the high-fidelity, low-loss response required in a (higher-offset) transmission band from
magnetic resonance.
26
CHAPTER 4 EXPERIMENTAL METHODS AND RESULTS
The primary research objective herein was to develop a design process that produced NIMs capable
of accurate frequency scaling and adequate bandwidth for microwave bands of applicable to space
communications. Essential properties this proposed material was that it had low-loss, uniform
transmission across the passband. Metamaterials with low insertion losses, large bandwidths, and
multi-frequency responses may be candidates for use in specialize microwave applications, such
as selected military and commercial telecommunications. However, in the past, one of the
challenges in developing these materials has been an unacceptably high mismatch between
theoretical and experimental responses. For example, errors in center frequency of NIM pass-band
filters are often unacceptably large for industrial microwave design that have very low error
tolerance. An important goal in undertaking this work was to achieve better theoreticalmeasurement agreement by combining the use of electromagnetic simulations as well as near-field
infrared (IR) imaging of the electromagnetic (EM) elements. The key engineering parameters
were: 1) accurate center frequency, 2) adequate pass-band for spread-spectrum modulation and 3)
low propagation loss within the passband. In addition, the purpose in creating several material
samples in different microwave bands was to validate frequency scaling of the design process.
Given these research objectives, NIMs were designed and tested for the several microwave bands
using the SSRR unit cell as the fundamental structural element. For each band, the unit cell was
designed in two stages; the first stage was single SSRR design and the second was array design
using finite element methods to design homogeneous slabs and tested. Next, slabs were
reconfigured as prisms, by cutting and rearranging the circuit boards. In prism configuration, the
IOR was measured by sweeping the receiver by rotation of 180 degrees about a platform holding
the material under test and subsequently repeating the process at incremental frequencies. The
design simulations and experimental measurements were then compared to evaluate the simulated
and measured results. In addition, bench-top measurements were conducted with an infrared
camera, to evaluate transmission in extended (2-D,3-D) dimensions for selected frequency ranges.
The utility of high-resolution to construct 3-D EM fields (as described in Sections 2 and 5) was
very instructive in understanding the modes of propagation and variations between experimental
and theoretical results. Some these results have been reported elsewhere by the author and his
collaborators in Dr. Pinchuk’s research group. [24] [25]
Further, specific design constraints were introduced to improve homogeneity in bulk materials,
and a dielectric with a low tangential loss component was used to lower the insertion loss. The size
constraint of the unit cell was such that each dimension for both designs needed to be equal to or
less than one-fifth incident wavelength (l/5). The prism materials consisted of 120 equally spaced
boards. The material slab for the K-band configuration were designed to have dimensions
exceeding 10 times the wavelength (10l); these dimensions are summarized in Table 5. The model
was extended to arrays of unit cells in order to simulate transmission in NIM flat slabs and prisms.
These symmetry design constraints probably made the material less sensitive to changes in
orientation, and likely contributed to low insertion loss due to uniform homogeneity.
27
Band
X-Ku
K
SSRR Design Parameter (mm)
a
b
h
w
c
d
l
5.2 4.0 5.0 2.8 0.4 0.51 2.0
3.0 3.0 2.04 2.04 0.2 0.51 1.0
Table 5. Actual design parameters for unit cells
After failed attempts to machine circuit boards or chemically etch the boards in house, the
metamaterial boards were ultimately fabricated commercially. The circuit boards were
commercial boards were delivered with regular sized unit cells arranged in arrays. These arrays
were set up using 10 unit cells aligned with the propagation vector and 50 cells perpendicular to
propagation, as shown in Figure 15. The boards were configured into a virtual slab (with air gaps)
by inserting boards into a specially machined device. The slabs of material were characterized by
their transmission and reflection coefficients (S-parameters) in the near-field by holding all
components stationary on a bench and using two microwave feeds and a network analyzer. After
transmission testing the bulk material, the materials were cut along cell alignment to create prisms
to conduct transmission testing for refraction.
Figure 15. (left) Circuit board of 10x50 unit cells with S-shaped split-ring resonators; (top
right) X-Ku prism; (bottom right) K-band prism
These angle and frequency characterizations were conducted in an anechoic chamber at the U.S.
Air Force Academy (USAFA). Figure 16 shows the metamaterial being tested in an anechoic
chamber as well as benchtop. Both setups used an Agilent 8753ES network analyzer. In this
experiment, a tripod holds an antenna horn, standing on a disk platform capable of 360-degree
rotation; however, for this experiment it rotated about 180 degrees, keeping the material under test
inside an absorbing foam window. The material was kept stationary with respect to incident waves,
during the platform rotation. Transmission measurements were recorded over a 180-degree
rotation of the receiving antenna. The distance from the transmitter to the receiver was 9.14 meters,
ensuring that the far-field measurement conditions and the distance from material to receiving
28
antenna was 0.76 meters. The mechanical rotation of the microwave horn did not introduce any
significant measurement noise, but the platform had to recalibrated to zero degrees before each
session to ensure consistent results.
(a)
(b)
Figure 16. (a) the anechoic chamber (b) photo of the test setup
A typical data set was created by the 180-degree rotation as it appears in Figure 17, which shows
an obvious peak power point. The experimental setup for this example is an unmodulated
continuous wave signal, showing a significant differential power near the peak power of about 340
degrees or negative 20 degrees.
29
However, many data sets do not show such a clear peak in the log scale. For that reason,
subsequent data sets are converted from log scale and plotted on a linear scale. Note that the power
scale in this instance is 5 dBm per division. The peak power indicated by the arrow in Figure 17
is roughly 10 dBm above the rest of the angular measurements. Given this clear differential at a
negative angle, the material under test is deemed a negative index material for that frequency.
While this example seems straightforward, there are many other cases using this material at other
frequencies where the interpretation may be ambiguous.
Figure 17. Log plot of power through a NIM prism
30
4.1 Experimental and theoretical results in X-Ku bands
Figure 18 is a transmission plot showing far-field experimental and theoretical transmission
spectra (S21), vertically polarized incident electromagnetic wave through the uniform slab of
metamaterial. Within the target frequency (12-13 GHz), the slab of metamaterial had a high
transmission band with less than 5 dB attenuation. The HFSS simulation results compared fairly
well with the experimental transmission spectra, with the transmission band around 11.5-13 GHz.
The model and experimental results of the pass-band indicate fairly good agreement, although the
model has spikes near 11.2 GHz that were not detected in this measurement. The peak
transmission in the model is about 12.4 GHz, while the transmission measurement was about 12.5
GHz, with a smooth pass band approximately 11.5 to 13.2 GHz, or nearly 1.7 bandwidth, using a
3 dB cut-off rule. The assessment is somewhat hampered by the experimenter choice not to
average (smooth) the spectrum.
Figure 18. Comparison of the transmission measurements MTM (red) and HFSS
theoretical simulation (black) through the X-Ku slab
31
After measurements of the transmission through the slab, several stair-step prisms were modeled
to estimate the IOR for selected frequencies. Accordingly, the X-Ku uniform slab received cuts
to prove directly the negative refraction of a plane electromagnetic wave transmitting through the
prism. An example of a simulation of transmission through the prism is shown in Figure 19 with
a diagram of the prism.
Figure 19. An HFSS transmission simulation (left) at 12.5 GHz through an extended
material array transmission through a prism (right) is shown.
32
A prism was constructed and was inserted into a foam widow and held stationary while the receiver
was rotated. To create the result over the X-Ku band, the measurement was repeated over many
frequencies to create data that differentiated frequency and refractive properties, as shown in
Figure 20. In this figure, negative angles correspond to negative refractive indices.
Figure 20. Transmission measurements by frequency and angle
33
In addition, Figure 21 shows the experimentally measured power of the electromagnetic wave
refracted at the back surface of the prism at different angles relative to the surface. The
measurements were performed using a range of X-Ku frequencies. The most obvious negative
refraction occurred between 12.5 and 15 GHz. For instance, at 13.75 GHz, the data indicated a
peak power at about -22o, as shown below.
-7
5
1375 MHz
x 10
4.5
4
Power mW
3.5
3
2.5
2
1.5
1
0.5
0
-80
-60
-40
-20
0
20
Degrees
40
60
80
100
Figure 21. Transmission measurements as a function of the angle due to a vertically
polarized electromagnetic wave (13.75 GHz) through the prism
Data generated from several frequency plots can generate a plot, as Figure 22 shows the negative
IOR of the X-Ku prism calculated. For instance, the IOR calculated to be about -3 near the
frequency of 13.5 GHz with a negative index of various values in excess of a 2-GHz passband.
Furthermore, the insertion loss was 0.42 dB/cell, which was comparable to the previous SSRR
designs. The data indicate only slight transmission loss for the X-Ku prototype. The positive
indices below frequencies of 12.5 GHz are indicated by the blue line with pronounced peak prior
to sign transition; the negative indices are indicated by the red line.
34
5
4
3
Index of Refraction
2
1
0
-1
-2
-3
-4
-5
8
9
10
11
12
13
14
15
GHz
Figure 22. Experimentally determined indices of refraction for the X-Ku band (blue line is
positive indices and the red line is negative)
35
4.2 Experimental and theoretical results in K band materials
Having evaluated the X-Ku band material in the first section of this chapter, the remainder of this
section will evaluate the result of the K-band material. The design resonance frequency of 19-21
GHz was selected for this materials and was considered as a candidate for a wideband frequencyhopped modulation. As claimed earlier, a modest change for this prototype to enhance symmetry
at the unit cell level. The results due to the changes were encouraging, indicating improved
agreement between simulated and experimental results based upon slab configuration as displayed
in Figure 23. The design center frequency of 20 GHz pass-band show good agreement, very likely
sufficient to satisfy most industry standards.
0
HFSS Model
MTM
-5
-10
dB
-15
-20
-25
-30
-35
-40
18
19
20
21
22
23
24
25
26
27
GHz
Figure 23. Simulation (black) and experimental (red) transmission through K-band
Again, data sets are combined to construct a plot (Figure 24) to show a predominantly positive
index material for frequencies between 19-21 GHz. At lower frequencies, the power that was
transmitted through positive angles and then shifted to negative angles at higher frequencies when
the transmission properties became predominantly negative-index.
36
Figure 24. Transmission measurements as a function of the frequency and angle
Figure 25 shows the transmission measurements at 21 GHz (K-band). At 21 GHz, this plot points
to highest transmission near -40°. The transmission loss was measured at 0.285 dB/cell, which
represents an unprecedented value for this kind of material. This modest improvement in
transmission may be attributed to the increased homogeneity due to the unit cell symmetry of the
design, but further analysis is needed to confirm this assumption.
37
-10
K/Ka @ 21 GHz
x 10
2.5
mW mW
Power
2
1.5
1
0.5
0
-100
-80
-60
-40
-20
0
Degrees
20
40
60
80
100
Figure 25. Transmission as a function of the angle (21 GHz, K-band) by the metamaterial
prism (peak at -40 degree)
Figure 26 shows the index values that were measured for a prism in K band. For instance, at 19.75
GHz, the K-band metamaterial yields  ≅ −1.8. In the figure, the positive IOR in this dataset is
on the lower frequency side of the center frequency, while the negative IOR is on the high
frequency side.
38
Figure 26. Experimentally determined IOR of the K band metamaterial: positive indices in
blue; negative indices in red
39
CHAPTER 5 INFRARED IMAGING FOR NEAR-FIELD ANALYSIS
As described in the previous chapter, direct microwave methods are the primary means for
measurement; however, an extended model for transmission is useful to analyze these materials in
three dimensions (3-D). For instance, in order to measure the transmission patterns through
gradients or prisms, as well as smooth surfaces, alternative methods were evaluated. In this
chapter, infrared (IR) imaging is explored as a complementary measurement technique to represent
the transmitted microwave energy through a prism.
Infrared imaging is an established method for conducting electromagnetic field research in extend
spatial dimensions. [26] However, its application in conjunction with metamaterials research has
not been explored prior to this investigation, some of which was reported previously in a
conference report by the author and his collaborators. [27] This technique requires power levels
sufficient to create thermal contrasts on heat-sensitive films, usually a few Watts at a short
distance. As the field energy converts to thermal contrast, an image on the film represents the EM
field strength. As a result, an image precisely maps the field in higher dimensions.
The motivation for imaging is to facilitate the analysis of spatial propagation in 3-D. As described
previously, modern IR cameras can translate electric field (units of V/m) to temperature (units of
degrees Celsius). As a result, high-spatial resolution measurements are possible due to the dense
detector elements in the camera’s focal plane array. Digital IR cameras implement analog-todigital converters to calibrate the temperature sensitivity yielding a very accurate digital images
that correspond to the electric field. In this experiment, the emitter provides power of a few Watts
which is generally sufficient to transmit through the material under test. Of course the transmission
leaving the material is a function of the thickness; to ensure accuracy, it is important to ensure that
the exposed film and the field of view are carefully aligned.
In order to evaluate IR imagery of metamaterials in irregular shapes (prisms), one must understand
Smith’s analysis [28] that explained an effect referred to as ‘enhanced diffraction’, was observed
during experiments using NIMs with stair-step surfaces. This effect does not need to be considered
for general applications (NIM slabs), only for experimental research when these special gradients
or prisms are used in determining the IOR. When appropriate for research, this is an adaptation of
Snell’s law for stair-step forms of the material. In Eq. 30, the term (⁄ ) is significant only when
it is near the magnitude as the other term:  sin( ); in fact, enhanced diffraction might occur
frequently in prism configurations. Unlike the other terms in Eq. 30, the additional term (⁄ ) is
not angle-dependent and thus can be eliminated when appropriate by careful discrimination.
sin() =


+  sin( ) (30)
Since enhanced diffraction was observed but not reported in Smith’s earlier paper [6] , it was not
well understood until later research. Due to this variable in the measurements, it should be stated
that the data analysis may not to be straightforward.
40
The use of IR imaging in NIM research is indeed novel, necessitating innovative analysis for a
deeper understanding that the general application. To estimate the refractive index using imaging
techniques, the array elements are assigned image rows and columns: Image(x,y) to a source
(microwave horn emitter with calibrated power). In this way, a beam may be outlined using the
set the (half-power) 3 dB point. For each image, calculation of the near-field IOR ( ) tracks this
procedure: (1) calibration: anchor the 3 dB point to image by emitter only (2) introduce the material
and mark the transmission peak, (3) measure the offset from center (primary beam), (4) convert
the pixel values to radial units, and (5) calculate  by Eq. 31.
 = sin( ) /sin( ) (31)
After testing the transmission with the uniform slabs, the boards were cut into stair-step prisms to
test the refraction of the electromagnetic waves as shown in Figure 27. Since the material is
constructed with discrete crystal-like cells, the prism yields a stair-step.
Figure 27. The NIM prism and diagram indicating step sizes
By Smiths’ description, when the stair step dimension (d) is much less that the incident wavelength
(  ≪  ) in Eq. 30, excessive diffracted energy should not be expected. However, if the value of
d is nearly the same as the wavelength of incident transmission (~), enhanced diffraction is
expected.
In this design (Figure 27), the prism was constructed by 8 steps (d), made of 15 boards, each board
having 2 mm separation, creating a step (d) of 30 mm. When the wavelength is between 25-20
mm, resulting in dimension d and wavelength are similar in scale. This means that for this
particular prism prototype in the 12-15 GHz range enhanced diffraction’ is expected. When
enhanced diffraction is present, the corresponding lobe should be considered an artifact in the
measurement; meaning that in the analysis, the additional lobe should be neglected in the analysis.
41
A bench-top experimental setup in the near-field through a metamaterial prism is shown in Figure
28. To provide energy, a signal generator is connected to an amplifier with a steady-state source
of 2 Watts with a center frequency at 12.5 GHz. A commercial grade mid-wave IR camera is
located at a distance shown to capture the field of view of exposed Kapton film. The material is
mounted on a transparent Styrofoam, which is transparent in the microwave. The (Merlin-Indigo
brand) camera has an air-cooled detector array using Indium antimonide (InSb) detectors. The
detectors have a heat sensitivity of 25 milliKelvin (mK) based on high-resolution analog-to-digital
converter and a nominal frame rate [29]
Figure 28. Laboratory setup for IR imaging of NIMs (drawing is not to scale)
As mentioned previously, to calibrate a baseline with the emitter horn (no material in transmission
path), an IR image was taken for a selected power. The calibrated image is important because
some initial image process is required before material testing, because most IR detector arrays
have elements can fail. These failures increase over time of operation for several reasons, resulting
in non-responsive pixels. To improve the array, a common image processing procedure is the
application of a 2-D median spatial filter to remove the non-responsive pixel values. In this case,
a 7x7 element median filter was employed, as shown in Figure 29.
42
Figure 29 Calibrated and processed IR image of the microwave beam of 2 Watts
Having calibrated an image for a selected power level, the material under test was introduced and
power was emitted. Consequently, thermal responsive with dynamic range corresponding to 2126 degrees as in Figure 30 (a) and (b) contour. In the figure, the two side-by-side images illustrate
the primary (high intensity) and secondary (low intensity) lobes. The plot’s temperature legend
indicates that the peak temperature (26 degrees) in the high intensity lobe while the second-order
lobe peak is lower by about 2 degrees. In the prism material, primary lobe corresponds to
refractive response and the secondary lobe to and diffractive response. [28] For the analysis of
refraction, the diffractive (low-intensity) lobe is excluded.
Figure 30. High-resolution IR image (a) and (b) contour map
43
By extension, an image sequence may be employed to construct 3-D representation. This is
possible equally spacing images to create an image stack or data cube. This data format is useful
in adding a third dimension. By this process, detailed analysis can be performed to compare the
theory and experimental results. For example, Figure 31 illustrates offset (1 cm increments)
images in thermal steady state laboratory conditions. A thermal contrast of 5.24oC produced by a
2 Watt emitter over an area of 12 in2 covered by film. In this format, data can be reviewed to
detect trends which imply through analysis any spatially dependent trends in the refractive indices.
Figure 31. IR Images at incremental distances of 1-4 cm
In addition, Figure 32 shows images corresponding to temperature range of 21-26oC through the
material in which there are two lobes. Evidently the thermal contrast apparent in these IR images
is about 5oC; which was comparable to the expected value of 5.24oC. Figure 32 also shows an IR
image the simulation to illustrate the fidelity of theoretical model and measured results with
regards to the lobes from the prism surface. This figure maps the electric field units (~1200-3500
V/m) to a temperature range of 21-26 degrees. There might be several reasons for the small errors
in the model versus the measured results, such as unknown propagation losses, but both
demonstrated the expected relative intensities. Accordingly, this new type of validation for NIM
simulation and measurement was possible.
44
Figure 32. Simulated lobes (top) and measured lobes (bottom) for same
configuration
With a calibrated image from the emitter source and the same with NIM prism image,
measurements contribute to a novel analysis for the of the IOR. As described previously, the
calibrated image maps the microwave source’s beam pattern. Next, as in Figure 33, a data image
slide of the NIM materials result and may be compared to the baseline.
Figure 33. Calibrated baseline (right) and the NIM transmission (left)
45
Following this procedure, two image slices (pixel rows) are extracted and superimposed on a
common plot. The calibrated image is used to estimate the and the beam width, bounded by the 3 dB metric. Next, this estimate is converted to pixel equivalent terms using the image centerline
as a reference. That is, the beam width can be compared to the image to match (as closely as
possible) with the one-half power points or (1/√2 ) pixel amplitude. In Figure 34, this match is
shown at 22 degrees. Given that data, it a pixel offset may be calculated as radial units and the
primary or first-order lobe is estimated which is 34 degrees in this case and that value is used in
Eq. 5. Here, the magnitude is sin(34/180)/ sin(8.1/180) = 4.0. The signed value is of – 4.0
given the negative side of the image, with respect to the IOR.
Figure 34. Image slide showing 1st and 2nd order lobes
The analysis described previously is the first step in a larger process. Images created by thermal
contrast are not expected to be perfectly symmetrical. Some variation is likely due to many
considerations; still, some quantifying data is desired to understand the range of the fluctuations.
To expand the limited analysis to represent the entire image, a family of curves may be
appropriate. This approach was executed to create Figure 35. Here, several values for refraction
ranging from: -3.8 to -4.5, as shown in Figure 35. In fact, the direct measurements for 12.5 GHz,
46
as seen in Figure 35(b) is ~ 4.2, is comparable to far-field measurements using the same prism.
Figure 35. (a) Family of curves by near-field analysis showing the IOR (b) IOR by direct
far-field methods (same material)
One may conclude that the method by near-field IR image-based methods and those by far-field
methods are in good agreement. In addition, the imaging method offers provides high spatial
resolution to construct 3-D data sets.
These 3-D data sets are complementary to the direct 1-D measurements that offer precise insertion
loss measurement for transmission, reflection, and phase. Further, direct methods are inherently
very low power while thermal contrast methods require considerable more power. In the closest
IR frames (bottom, incremental distances), the dominant heat profile indicated a negative
refraction. When combined with direct far-field microwave measurements, IR imaging enhances
the investigator’s ability to compare EM fields in high-resolution 3-D to verify negative indices
within frequency ranges.
Again, the research is designed to analyze uniform materials not anomalies introduced the irregular
stepping in the prism configuration that is not an organic property of the material itself that would
be evident in uniform slabs.
47
CHAPTER 6 POLARIZATION MEASUREMENTS
In order to assess the polarization effects of the metamaterials, two rotational measurement
techniques were implemented. Those techniques are axial and lateral rotation. A subset of these
results have been reported by collaborators in Dr. Pinchuk’s research group, including the author.
[30] [31] Figure 36 illustrated the setup for the axial rotation. First, a single-axis (1-D), threelayer material was mounted on a Styrofoam disk, which was mounted on a Styrofoam block with
a groove of sufficient width to permit full rotation of the disk. Transmit and receive microwave
horns (ETS 3106-09) were separated by a 25 cm distance.
Figure 36. Experimental set up for axial rotation
In axial rotation, the 3-cell material slab and the microwave horns are aligned in the same (vertical)
E-field orientation. Consequently, the material’s effective surface area does not change in axial
rotation. In the experiment, a quarter rotation beginning at the 0-degrees position (vertical E-field
alignment) through the 90-degree position in equal increments to measure any distortion of
transmission due to polarization misalignment; these results are seen in Figure 37.
48
Figure 37. Transmission of 3-cell thick single axis as a function of axial rotation angle
When the material is closely aligned with E-field of transmission at reference (0 degree) position
rotation, (red in Figure 37) design transmission is closely approximated. Then as the material is
axially rotated to 22.5 degrees (green in Figure 37), a small but noticeable change occurs,
especially outside the passband, while a negligible change is observed in the passband. However,
in the next step of a 45-degree (cyan in Figure 37) axial rotation, a distinct reduction in both
passband as well as elsewhere is observed. Beyond 45 degrees of rotation, the final two
transmission measurements show that the material is nearly transparent at 67.5 degrees (blue) and
virtually transparent at a 90-degree rotation (black). These transmission (S21) results are
consistent with the theory. When the materials slab is misaligned by 90 degrees with the material,
the unit cell does not behave as a metamaterial; that is, the design transmission is irresponsive
because the unit cell must be closely aligned with the E-field to be effective. The fact is that the
materials tolerate misalignment quite well up to a 20-degree rotation.
Lateral rotation involves a rotation of the slab so that it is no longer normal to the plane; as a
consequence, the exposed surface area declines with each increment. The measurements were
performed with an ETS 3164-05 microwave horns and the material under test was mounted in a
Styrofoam as shown in Figure 38. The radiating horn was close enough to ensure that the energy
passed through the material.
49
Figure 38. Experimental Set up for lateral rotation
The transmit and receive antennas were placed at a distance of 25 cm apart, in same orientation
keeping a linearly polarized electromagnetic signal, as shown in Figure 39. The transmission
through air (no material) was also included as a baseline.
5
0
15
30
45
60
Air
0
-5
dB
-10
-15
-20
-25
-30
18
19
20
21
22
23
24
25
26
27
GHz
Figure 39. Lateral rotation of the 3-cell thick, single axis metamaterial
50
Next, the transmission was measured through the dual-axis (2-D) SSRR metamaterials at
increments in axial and lateral rotation. The material under test was rotated in 45-degree
increments, as shown in Figure 40.
Figure 40. A photograph 2-D metamaterial slab (inset) for angle-dependent measurements
The goal of the 2-D design was to reduce the sensitivity of the transmission to orientation; that is,
the interlocking cells provide a matrix of the material that creases a response similar to the original
orientation, where the E-field is perfectly aligned with the unit cells. As shown in Figure 41, the
1-D material and the 2-D show transmission in axial rotation. The 1-D material performs as
designed with perfect alignment (0 degrees), while it suffers degradation as the material is rotated
within the E-Field, approaching nearly flat transmission across the frequency span at 90 degrees.
In contrast, the 2-D (interlocking cells) material has a quite stable transmission profile across the
frequency span, indicating insensitivity to orientation.
51
Figure 41. Illustration of 1-D (left) versus 2-D design (right) transmission in rotation
52
CHAPTER 7 CELL THICKNESS, BANDWIDTH, AND TRANSMISSION LOSS
Material thickness is another consideration in the bulk design of metamaterials. Thickness must
be taking in increments of whole cells, since partial cells are not functional units. As previously
mentioned, a thickness of 3 cells has been demonstrated to retain the design characteristics in the
bulk material quite well. As the material increases in thickness the pass-band characteristics
become more pronounced: the pass-band remains roughly the same, but the out-of-band rejection
is much stronger.
In applications where the bulk material should be a thin as possible, the out-of-band rejection may
have to tolerate some undesired transmission on the fringes of the 3 dB point. However, when the
designer can afford the thicker material, greater control can be had. Figure 42 illustrates this
transmission behavior using a NIM designed for Ku band transmission. The thicker material (6cells) demonstrates a steeper roll off, and accordingly, it is possible to design materials
transmission characteristics in bulk by selecting the number of cells appropriate to achieve the
materials’ ability to selectively filter electromagnetic waves by frequency.
0
3 layer
6 layer
-5
dB
-10
-15
-20
-25
-30
12
12.5
13
13.5
14
14.5
GHz
15
15.5
16
16.5
17
Figure 42. Metamaterial transmission comparing cell layers of identical material
The K-band design required the most stringent constraints due to the symmetry of its unit cell to
enhance homogeneity. Design constraints were imposed to achieve acceptable homogeneity. Since
the raw materials in both the X/Ku and K-band were the same in Table 6, the unit cell symmetry
53
(in the K-band design), which could be a possible explanation for more predictable agreement
between simulation and test results. In addition, symmetrical unit cells lend to ease of fabrication.
Band
X-Ku
K
Res. Freq.
(GHz)
12.5
20
Insertion Loss
dB/cm
dB/cell
1.69
0.422
0.95
0.285
Table 6. Prototype pass-bands with corresponding insertion loss
As documented previously, the transmission loss through metamaterials is represented in units of
dB/cell or dB/cm. As shown in Figure 43, the trend in signal transmission loss in negative index
materials over the past 15 years indicates using units of dB/cm. The earliest (Smith, 2000) [6] data
point evidences a relatively high loss at 8 dB/cm with some improvement reported in 2003. During
this period, most NIMs employed variations of split ring resonators with rods which are associated
with relatively high loss. Chen [22] showed that, with the elimination of rods as in the first
instantiation of the SSRR, much lower loss could be realized. In this research, even lower loss was
realized with symmetric SSRR. The correlation of lower loss may be due to greater homogeneity
of the materials. Lee [24] reported slightly better results than Chen ( -1.75 vs. -0.95 dB); to date,
this is the lowest loss of reported in a NIM.
Figure 43. Trend in transmission loss in NIMs over the past 15 years
54
CHAPTER 8 PROPOSED APPLICATIONS
When Veselago proposed the theory for negative index materials, he did not describe any specific
potential application, but only derived the physics and its alignment with Maxwell’s equation. It
was much later, after the realization of NIMs, that serious thought was given to any practical
benefits to technology.
8.1 Radome materials for Wideband/spread-spectrum signals
Another possible application of metamaterials is the design of specialty radomes. The idea has
been reported previously [32] [33]; however, other researchers have focused on radomes for
narrow bandwidths. In order to achieve ultra-wideband radomes, novel approaches in materials
engineering must be considered. In general, the purpose of the radome is to provide a physical
barrier between the environment and an antenna. Structurally, the radome should shield the
antenna and associated electronics from the atmosphere, preventing damage from wind, humidity,
gaseous pollution, and particulates. Ideally, the radome should not attenuate or distort the signal
being transmitted; the incident power, bandwidth, and modulation should not be altered in an ideal
radome. Of course, in practice, an ideal performance is rarely achieved, and the design engineer
must understand the performance trades in the design problem.
Currently, there are many acceptable conventional materials and methods to achieve very good
electrical characteristics in narrow-band modulated signals (≤ 10 KHz in bandwidth) and ≤ 10
GHz carrier frequency. These materials are appropriate for most commercial modulation schemes
and frequency bands; however, for spread-spectrum modulation, specialty radome materials may
be required to reach near-optimal performance.
Originally designed for the military, spread-spectrum modulation is used by military and
commercial space and terrestrial-based systems. Spread-spectrum modulation falls into two
categories: direct-sequence as well as frequency-hopping. The US Global Positioning System
(GPS) is an example of direct-sequence spread spectrum, and its bandwidth occupied 2-20 MHz,
depending on the mode: civilian or precise. Indeed, this is a wideband signal, but by comparison,
frequency-hopping spread spectrum in the extremely high frequency band may occupy 1-2 GHz
of bandwidth, of magnitudes greater than GPS. [34]
The challenge of developing a radome material that provides high and uniform transmission at
extremely high frequencies (>20 GHz) is formidable. Radome materials for EHF transmission
exist, but improvement in transmission and uniformity are highly desired to yield robust space-toearth satellite links. The problem is illustrated by simulating transmission using commercially
available software, based upon a popular textbook on radome engineering; see Figure 44. [35] The
simulations offer several design choices for dielectric, layers, thickness, etc., and the frequency
response band; over-laid is a curve of radome loss based upon a 4-layer NIM material. Clearly,
the signal loss through the NIM material is much less and it is more uniform than the conventional
material of comparable thickness as shown.
55
Figure 44. Conventional materials compared NIM designed for 20-22 GHz
These simulations show the non-uniform (damped oscillating) transmission property over
wideband, which varies as much as 3-4 dB peak to peak. Again, this behavior may not create a
problem for signal with narrowband modulation techniques; however, it does in particular for the
frequency-hopping spread spectrum, where the signal is coded to ‘hop’ over the entire band with
a pseudo-random pattern. Conceivably, the two time-consecutive hops could exhibit a 3-4 dB
difference in signal strength, because those are in different portions on the band. Accordingly,
appreciable signal amplitude distortion will occur with uncertain impact on bit error-rates,
especially in stressed electromagnetic environments, including ionic scintillation, heavy rain, and
adversarial interference. Typical satellite link budgets allocate no more than 1-2 dB for radome
losses [36] for high-performance satellite communication systems, affording very little margin
when compounded factors impose highly stressed scenarios. For this reason, alternatives such as
metamaterial to conventional radome materials is to be explored. In this research, very wide,
uniform, and low-loss pass-band materials have been designed and tested. In fact, the electrical
properties were designed to match the downlinks of operational satellites. In future research, the
electrical and mechanical properties must be considered to determine whether or not a NIM
candidate might fully satisfy or complement the existing radome materials. Figure 45 is a
computer-aided drawing of the SSRR radome material, with equally spaced unit cells and singlecell depth. Such a design would be quite thin and light weight, but layered unit cells might be
necessary to provide the out-of-band rejection required. Light weight but stout foams with
emissivity comparable to space could be used as a fill material.
.
56
Figure 45. Single layer design
8.2 Microwave vertical/horizontal polarizer
This section proposes a low-cost material which may improve cross-polarization isolation for
commercial communication links.
By convention, commercial space-based microwave
communications links designate a vertically polarized signal as the uplink and a horizontally
polarized signal as the down link. A ground antenna (usually) transmits and receives on the same
antenna dish, requiring high cross-polarization isolation to achieve robust operation. Depending
on its mission, an antenna may be mobile or stationary; whenever an antenna is repositioned
(relocated or disturbed), the operator is required to telephone the satellite operations center to
verify that the proper transmit power and signal polarization meets the contracted specifications.
These power and polarization checks usually require an iterative process in real time, beginning
with cross polarization separation under which 28-30 dB is generally considered acceptable, but
higher that 30 dB is desired to avoid cross-polar interference to neighboring microwave leased
spectrum. Typically, this process may take two people at the ground station an hour or more.
If cross-polarization compliance slips, the satellite operations center may require the ground
antenna to recalibrate before transmitting. This process of recalibration is required of smaller
portable antennas or non-penetrating rooftop units, where even slight wind conditions may affect
precise positioning. High cross-polarization separation [37] has been achieved, even with less
expensive antenna equipment, but the additional process is labor intensive and manually tedious.
In order to achieve high cross-polar (V-H) separation on a less expensive antenna, one possibility
might be to insert a patch two-layer material of negative index material; the purpose of this patch
would be to reduce cross-polar interference between vertical (uplink) and horizontal (downlink)
signals. Preliminary results with this design indicate that an additional 3-4 dB in isolation is
possible, if installed properly; that might effectively translate to 35-36 dB for routine calibration
for the even less expensive 1.2 m and 2.4 m (diameter) in Ku band; see Figure 46 and 47.
57
The benefit of higher isolation is that the operator might transmit higher power without interfering
with his spectral neighbors. Higher power, when necessary, might allow operations during rain;
the reliability of maintaining space link (transmission/reception) is often affected by weather
conditions. Many Ku band communications are narrow-band (bandwidth ~ 100 kHz), but some
Ku modulation used direct-sequence spread spectrum (DSSS) modulation for security; the DSSS
modulation may require up to 40 MHz of bandwidth.
Figure 46. Ku antenna 2.4 meter system
58
H - polarized Low Noise Amplifier with
Down
- Convertor Local Oscillator
V - polarized Block Up Convertor
-
NIM
with Local Oscillator
Common waveguide
Figure 47. Ku antenna collector with up/down converter with NIM and exaggerated board
spacing overlaid to show collector
Initially, the concept could be tested in line with conventional antenna feeds, but if proved
successful, fully integrated NIM shields are envisioned. Of course, many space communications
are circularly polarized signals; this concept is not proposed for that class of communications. In
addition, the cross-pol isolation offers no benefit for the receiving-only antenna.
8.3 Other emerging applications
In addition to academic research, considerable private commercial research is being conducted in
microwave metamaterials.
Since industrial research is often protected by proprietary
nondisclosure, the performance claims made may not be fully understood by the larger engineering
community. However, a few commercial ventures have offered a glimpse into the overall
objectives for their development, while keeping the development process an industrial secret.
The claims of commercial research, although not fully disclosed, indicate that antenna size may
be reduced considerably through the use of metamaterial covers to focus radiation on active
elements. Further, antenna beams can be shaped for additional gain, as needed for agile tracking,
which is needed in highly dynamic scenarios when both transmitting and receiving antennas are
moving.
Other examples of metamaterial application to antennas are high data throughput in ultrawidebands; in addition, low-profile (conformal) high-gain and beam-steering antennas have been
demonstrated recently. [38] [39] The antenna gains reported from planar prototypes vary (12-80
dBi), which is comparable to dish antennas, which are much larger and more difficult to integrate.
59
[40] Furthermore, convention antenna arrays draw high power and have suffered reliability
problems because of the numerous components that must be maintained. Also, through improved
electrical compatibility, antenna ‘farms’ on space and sea platforms may be integrated into smaller
packages without introducing spectrum interference. Table 7 contains a limited survey of potential
applications, but this survey is only a fraction of those being considered.
Potential Application
Performance Claim
Key technical enabler
Radome materials for wideband
microwave signals
Uniform transmission over GHz
bandwidth at very high transmit
frequencies; unlike conventional
composite materials with oscillating
transmission patterns.
Cross-polarization isolation in excess
of 28 decibels is needed to maintain a
constant, noninterfering link with
satellites in geostationary orbits.
Analysis shows that a modest gain of
3+ decibels may reduce the need for
regular realignment.
Reduced size, mass and bulk on
antenna platforms, especially those that
require aerodynamic surfaces. Gains in
excess of 12 dB have been reported.
Demonstrations of metamaterials show
how radiation from multiple sources
can be better controlled to substantially
reduce interference from emitters in
very confined spaces, which may
benefit the retrofit of more capable
replacement antennas into the existing
platforms.
Low-power high beam detection and
tracking, largely in the commercial
domain, where transmitters and
receivers are dynamic.
Components in transmission lines may
be alternatives to current devices.
Control reflection on metallic surfaces
for the purpose of reducing multipath.
Loaded transmission lines to enhance
bandwidth.
Using symmetric 3-D unit cells to
create materials insensitive to
polarization and incident radiation.
Panel Vertical or Horizontal
polarizers for transmit and
receiver dishes for satellite
transmission to enhance isolation
Conformal, high-gain antennas
Electromagnetic compatibility
Agile beam forming
Dispersion / Phase tuning
Ground planes
Loaded lines
1-D panels of NIMs provide
enhance alignment to either
vertical or horizontal polarization
on separate transmit and receive
dishes.
Negative Index Materials over
patch antennas; for example, direct
radiation, much like a dish, but
without the bulk.
Dielectric metamaterials show
promise in directing the flow of
energy.
Forward
creation.
and
backward
beam
High frequency components in line
with transmission wires.
Precise reflection (S11) control.
Cell thickness and other variables.
Table 7. Limited Survey of Potential Applications
60
CONCLUSIONS
The publication of Veselago’s theory for negative index metamaterials, which has been cited
thousands of times, preceded its realization by several decades. Indeed, at the time, Veselago did
not know whether or not the theory could be proven, since a clear pathway to fabricate and test
those materials was not identified in the late 1960s. Derived from the fundamental laws of
electromagnetism, Veselago’s proposed materials were expected to have unusual electromagnetic
properties. Based upon those unrealized properties, new applications that were imagined but not
be verified in his original paper.
Three decades later, Pendry took fresh interest in the theory and furthered the investigation with
remarkable insight by designing devices such as the split ring resonator capable of producing
negative electromagnetic parameters. Pendry’s published designs enabled the realization of
negative index materials by inventing models for devices that could produce simultaneous negative
permeability and permittivity. These models were used by Smith and his team, who actually
constructed the first negative index metamaterial, attracting considerable excitement. Verification
of Smith’s microwave-domain experiments by several research groups corroborated that the
phenomenon of negative refraction was real, but lingering questions regarding its utility remained.
For example, could the material be engineered to satisfy the desired electromagnetic properties as
well as the mechanical requirements for an integrated system or components?
Subsequent to Smith’s initial research, studies indicated that transmission properties could be
modified through creative design of unit cell and extended 3-D slabs consisting of unit cell arrays.
Properly designed, bulk materials could be made effectively homogeneous when the unit cell was
designed much smaller than the wavelength: usually equal to ≤ /5 of the design incident wave.
However, achieving effective homogeneity for a particular design is a challenge that may require
several iterations of design. This research indicates that a minimum thickness of three unit cells
constitutes a material with properties consistent with design properties. Additional layers afford
steeper roll-off in a band-pass filter.
This research also addressed the deficiency in design fidelity in transmission bands which were
candidates for wideband signals. That is, a rapidly converging design method that results in precise
center frequency as well as uniform low-loss, pass-band appropriate for spread-spectrum
modulated signals. High-frequency (especially > 10GHz) conventional materials usually designed
for narrowband modulation are limited in this respect, creating the motivation to explore
alternative materials. Building upon previous work by Chen et al., a new design approach arose
based upon S-Shaped Split-Ring Resonators that incorporates symmetry of unit cells, and a rapid
convergence process resulted in unprecedented low-loss NIMs, with high confidence design
methodology.
The development of the design method was aided by the use of IR imagery, which afforded a
comparison between the direct RF measurements and indirect thermal measurements to estimate
the IOR. Calculations for IOR using both methods resulted in comparable estimates and also
provided insight into the ‘enhance diffraction’ effects produced by the stair-stepped prism. This
enhanced diffraction phenomenon was observed when transmitted through the prism, but was
initially not understood until further investigation revealed the cause. As a result, the enhanced
diffraction effects (occurring only in the prism configuration) were not included in the evaluation
of IOR. In addition, the imaging method offers greater insight in 3-Dimensions, revealing a
61
slightly non-uniform distribution. The dual method approach of taking direct microwave
measurements complemented by imagery was innovative in its application to microwave-domain
metamaterials.
Several material prototypes were created in this research to address the question of frequency
scaling, of which were reported in this paper. The 20 GHz center frequency design was the result
of a very good match and was complemented by very low transmission loss. The HFSS simulation
and the measured result were a good match. This model-experimental precision is attributed to a
systematic approach emphasizing unit cell symmetry that converges within only a few design
iterations, prior to implementing the final high-fidelity step of the finite element method. The
commercial simulation tool supports a host of available circuit board materials that can be selected
for testing. Once the circuit board components are fabricated, the final assembly was possible as
slabs, prisms, etc. The possible configurations may be suitable for many applications.
The criteria for layered cells depends on the desired properties, such as bandwidth and transmission
cumulative loss. In the case of radome applications, for example, a 3-cell layer seems to be a good
design rule to retain the design wideband transmission characteristics, without imposing
significant loss; in fact, this research produced unprecedented results in terms of lowest insertion
loss for high frequency and wideband materials. With high-fidelity design tools and methods, it
seems clear that metamaterials shall satisfy some engineering requirement that conventional
materials can not satisfy. For instance, metamaterials offer properties in impedance matching,
tunable transmission, wide bandwidth, etc., that are peculiar to negative permeability/permittivity
characteristics.
The microwave engineering community has just begun to explore the possibilities of improving
existing engineering trade spaces as these novel materials become available. It remains
speculation exactly where the most promising applications will be. It could be that metamaterials
fill a need only in niche applications or perhaps the impact might be broad in scope, including new
practices not yet envisioned. The application of metamaterials designed for radomes, especially
for high frequency spread-spectrum modulated signals, is intriguing because it fills a gap in
conventional materials.
In all of these potential applications, the availability of high-design fidelity design is an underlying
assumption. This research should contribute to the body of research, public and private, to advance
state of the art design in applying metamaterials microwave engineering. The key contributions
to design methodology are the recognition of unit cell symmetry in design fidelity and
homogeneity as well as extended dimensional analysis provided by IR imagery.
Important considerations for future research may include a more automated fabrication and testing
of metamaterials. Very likely, more can be learned as the pace of design, enhanced modeling, and
evaluation are accelerated. As a family of metamaterials is introduced to a new generation of
engineers, innovative approaches to solving the existing challenges are likely to emerge and made
commercially available with supporting design tools.
62
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65
APPENDIX
A.1
For this research, a modeling tool has been developed in MATLAB to design SSRR unit
cells for printed circuit boards. The MATLAB script contained within is a result initially of the
work of Mr. James Vedral (University of Colorado graduate student in physics) and later
modified to facilitate the use of key parameters and rapid convergence on the desired
transmission properties.
The script has been tailored after many iterations of trial and error to avoid redundancies to
provide the essential design parameters for the final modeling step using the HFSS finite element
method. The user is required to specify initial parameters for the design as indicated in the
following template in table 8.
Parameter
Description
Frequency (GHz)
Circuit Board thickness (m)
Resonant frequency
Variable by industry convention
(0.508 e-3 or 0.508 mm by default)
Circuit Board spacing (m)
1 mm by default
Copper trace resolution (epsilon)
User selection
Unit cell size in terms of wavelength
Whole number fraction of wavelength
Design tolerance (delta frequency GHz) Acceptable error
Table 8. Template for design inputs
In addition, Figure 48 illustrates a flowchart for the design method using the MATLAB script.
66
Figure 48. Flow chart for MATLAB code design tool
===========================================================
% This MATLAB script provides the user a simulation tool to
design a S-Shaped Split Ring Resonator corresponding to a design
frequency within given tolerances
% Prompt user input
str=input('Enter Frequency GHz: ','s')
freq = str2num(str);
%
Initial parameters
c=0.3250E-3;%Cu trace width (m)
epsilon = 0.001e-3; % incremental change
n=0;
wavelength_div=5; % cell size by wavelength fraction
vc=3.0e8; % speed of light (m/s)
GHz=1e9;
wavelength=vc/freq/GHz;
67
a=wavelength/wavelength_div;
b=a; % assign outer boarder symmetry
x=0.75
% fraction of inner boarder
h=x*a;
w=h; % inner SSRR boarder symmetry
L=1e-3; % board spacing
d=0.508e-3; %
er=3.65; %
10-30 GHz
%
board thickness
Dielectric constant
-- Rogers Duriod 4003C between
call the algorthigm to calcuate resonant frequency
[f]=met_v3(freq,er,c,a,b,h,w,L,d,wavelength_div)
delta=freq-f; % delta in frequ.
tolerance = 0.05; % in GHz
% Converge on tracing 'c' size
% call metamaterial fuction
while(abs(delta)>tolerance)
[f,Cm]=met_v3(freq,er,c,a,b,h,w,L,d,wavelength_div)
% Calculate the frequency difference
delta=freq-f;
if(delta>0)
c=c-epsilon;
else
c=c+epsilon;
end
n=n+1;
end
68
function [fest,Cm]=met_v3(freq,er,c,a,b,h,w,L,d,wavelength_div)
% Given S-shaped SRR parameters, this matlab file calculates the
theoretical %resonance frequency or frequencies. The center bar
may be adjusted to %achieve different volumes for FI and F2. A
setting of 0.5 means it is in
%the center and Fl=F2. The resonance frequency or frequencies
are given and
%a plot evaluating the permeability relation is output.
%Calculate resonant frequency given unit cell dimensions of an
S-SRR.
cond_s=0.5; %Cu Resistance
F12=((((a*b*L)-(h*w*L)))/(a*b*L));
center_bar=0.5;
% calc volune ratios F1 & F2 bases ib cebter bar and S-shape
dimensions
F1=(center_bar*(h*w*L))/(a*b*L);
F2=((1-center_bar)*(h*w*L))/(a*b*L);
%free space consts
epsilon_0=8.85E-12;
mu_0=1.257E-6;
%strip capacitances
iterate=1;
Cs=(er*epsilon_0*((h*c)/d))+(epsilon_0*((h*c)/(L-d)));
% Equate capacitance of mid / high low metal strips
Cm=Cs;
69
% Calculate Loop area
S=a*b;
%case when F1=F2.
wm0_squared=(1/(mu_0*F1*S))*((L/Cs)+((2*L)/Cm));
wm0=sqrt(wm0_squared);
fm0=wm0/(2*pi);
m=F1/F2;
n=Cm/Cs;
% resonance when F1=/=F2
wm01=sqrt((((m+1)*(n+1)+sqrt((m1)^(2)*(n^(2)+(2*n))+(m+1)^(2)))/(2*m*n))*(L/(mu_0*S*F2*Cs)));
fm01=wm01/(2*pi);
wm02=sqrt((((m+1)*(n+1)-sqrt((m1)^(2)*(n^(2)+(2*n))+(m+1)^(2)))/(2*m*n))*(L/(mu_0*S*F2*Cs)));
fm02=wm02/(2*pi);
diff_f0=((max(fm01,fm02)+2E9)-(min(fm01,fm02)));
num_samples=((max(fm01,fm02)+3E9)/1E9)/0.01;
step_size=0.01E9;
%create freq. points in vector for plotting.
for a=1:num_samples
freq_matrix(a,1)=((0)+a*step_size);
end
omega_matrix=freq_matrix.*(2*pi);
%eval permeability relation for each freq point.
70
A=omega_matrix.*mu_0.*S.*(F1^2+F2^2).*L;
B=(cond_s.*L)^2;
C=(omega_matrix.*mu_0.*S.*(F1+F2)-(2./
omega_matrix).*((L/Cs)+(L/Cm)))*cond_s*L;
if(F1~=F2)
mu_eff_num=(omega_matrix.*mu_0.*S).^2.*F2.*F1.*(F1+F2)(mu_0*S).*((F1^2+F2^2).*(L/Cs)+(F1-F2)^2.*(L/Cm))+(1i.*A);
mu_eff_den=(omega_matrix.*mu_0.*S).^2.*F1.*F2(mu_0.*S).*(F1+F2).*((L/Cs)+(L/Cm))+((1./((omega_matrix).^2)).*(
L/Cs)).*((L/Cs)+((S*L)./(Cm)))-B+(1i.*C);
end
if(F1==F2)
X=(omega_matrix.*mu_0.*F1.*S).^2.*(1(1./(omega_matrix.*mu_0.*F1.*S))*(L/Cs));
D=A./X;
E=B./X;
G=C./X;
mu_eff_num=F1;
mu_eff_den=1(1./((omega_matrix).^2.*mu_0.*F1.*S)).*((L/Cs)+((L)/Cm))E+(1i.*G);
end
mu_eff=1-(mu_eff_num./mu_eff_den);
mu_eff_real=real(mu_eff);
mu_eff_imag=abs(imag(mu_eff));
x=freq_matrix./1E9;
y=mu_eff_real./100;
71
figure(1);
plot(freq_matrix./1E9,mu_eff_real./100,'k','linewidth',2);
grid on;hold on;
H=[x,y];find(max(H));
m=max(H(:,2));[I]=find( H(:,2) == m );
fest=H(I,1);% frequency estimate
title('Real Part of Permeability from SSRR');
xlabel('Freq(Ghz)');
ylabel('Re(mu)');
hold on;grid on;
figure(2);
plot(freq_matrix ./1E9, mu_eff_imag./100,'r','linewidth',2);
title('Imaginary part of permeability from SSRR');
%axis([ 10 50 ]);
grid on;hold on;
xlabel('Freq (Ghz)'); grid on;
ylabel('Imag(mu)');
% Normalize to frequency units of GHz
x1=freq_matrix./1E9;
y1=mu_eff_real./100;
y2=mu_eff_imag./100;
%
return
A.2
Resolution of units in Frequency calculation for magnetic resonance in SSRR
This section tracks the consistency of units to ensure the equations used in the design algorithm
were properly used to arrive at units for frequency.
72



 = √
( +
)
  
1

 = √
( )

()(2 )() 
1

 = √
( )

(2 )() 
 = √
m
 2
(2 )()(  )
1
1
 = √(2 ) =  (Hz)
1
1
 = 2 √(2 ) =
1

(Hz)
The area of the unit cell is S and the Fraction (F) area of the half ring (ab/2) is a key design
parameter that determines the size of the unit cell. Higher frequency unit cells are correspondingly
smaller in size.
 =  ( )
=
 
 
(unit less)
In addition, the capacitance created ( top/bottom of S-shape and  middle of S-shape) by
matching metallic strips (h*c = metal area) separated by board thickness () of dielectric material
and by neighboring boards separated by free space at a distance ( − ) may be calculated.



−
 =  =     ( ) +   (
) (Farad) or ( /)
73
Wideband and Low-loss Negative Index Metamaterials
for Microwave Antenna Applications
by
David Allen Lee
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted
without fee provided that copies are not made or distributed for profit or commercial advantage and that
copies bear this notice and the full citation on the last page. To copy otherwise, to republish, to post on
servers or to redistribute to lists, requires prior specific permission and may require a fee
74
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